Symplectic invariants for parabolic orbits and cusp singularities of integrable systems with two degrees of freedom
Alexey Bolsinov Lorenzo Guglielmi Elena Kudryavtseva














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Symplectic invariants for parabolic orbits and cusp singularities]Symplectic invariants for parabolic orbits and cusp singularities of integrable systems with two degrees of freedom


Department of Mathematical Sciences,
 Loughborough University,
 LE11 3TU, UK  and  Faculty of Mechanics and Mathematics, Moscow State University, Moscow 119991, Russia
 A.Bolsinov@lboro.ac.uk 
SISSA, Via Bonomea, 265 – 34136 Trieste, ITALY
lgugliel@sissa.it
Faculty of Mechanics and Mathematics, Moscow State University, Moscow 119991, Russia
eakudr@mech.math.msu.su



[
    Alexey Bolsinov, Lorenzo Guglielmi and Elena Kudryavtseva
    December 30, 2023
=============================================================






We discuss normal forms and symplectic invariants of parabolic orbits and cuspidal tori in integrable Hamiltonian systems with two degrees of freedom.  Such singularities appear in many integrable systems in geometry and mathematical physics and can be considered as the simplest example of degenerate singularities. We also suggest some new techniques which apparently  can be used for studying symplectic invariants of degenerate singularities of more general type. 







§ INTRODUCTION
 


An integrable Hamiltonian system on a symplectic manifold (M^2n, Ω) is defined by n pairwise commuting functions F_1, …, F_n which are independent on M^2n almost everywhere.  We will consider the case  n=2 and denote such a pair of commuting functions by H and F  (H is usually considered as the Hamiltonian and F as an additional first integral).   Under the above assumptions,  on M^4  we can introduce the structure of a singular Lagrangian fibration whose fibers are, by definition, common level surfaces  ℒ_h,f = { H = h,  F=f},   (h,f)∈^2  (or their connected components).  We will assume that all the fibers are compact  (unless we study local properties of a system). The functions  H and F also define a Hamiltonian  ^2-action on M^4.

According to Liouville theorem,   regular compact connected fibers are 2-dimensional Lagrangian tori of dimension 2  which coincide with orbits of the ^2-action.    We say that a fiber ℒ_h,f is singular if it contains a singular point, i.e., a point P such that dH(P) and dF(P)  are linearly dependent.  Equivalently, we may say that  ℒ_h,f is singular if it contains an orbit of a non-maximal dimension, i.e., 1 or 0.  A general problem of the theory of singularities of integrable systems is to describe the topology of singular fibers and their saturated neighborhoods (similarly for singular orbits). Notice that the fact that F and H commute makes this theory rather different as compared to the classical  singularity theory for smooth maps.  

Saying “describe”  we may mean at least three different settings:  topological, smooth and symplectic.  For instance, saying that two given singularities   (points, orbits or fibers)  are symplectically equivalent  we mean the existence of a fiberwise symplectomorphism between their neighborhoods.  Throughout the paper, in addition we will assume that all the objects we are working with are real (or complex) analytic.


In this paper we discuss just one particular type of singularitites,  namely parabolic orbits and cuspidal tori (speaking informally,  a cuspidal torus is  a compact singular fiber that contains one parabolic orbit and no other singular points). 

Recall that typical  (non-degenerate)  singular orbits in integrable Hamiltonian systems can be of two different types: elliptic and hyperbolic.  
In integrable systems of two degrees of freedom, we may  very often  observe a  transition from elliptic to hyperbolic in a smooth one-parameter family of singular orbits.  At the very moment of transition,  the orbit becomes degenerate and of parabolic type.  This scenario is rather natural and  parabolic can be viewed  as the simplest possible type of degenerate singularities.
 
 Another important property of parabolic orbits is their stability  under small integrable perturbations (this follows from <cit.>).  This is one of the reasons why such orbits can be observed in many examples  of integrable Hamiltonian systems:    Kovalevskaya top <cit.>, other integrable cases in rigid body dynamics including Steklov case, Clebsch case, Goryachev–Chaplygin–Sretenskii case, Zhukovskii case, Rubanovskii case and Manakov top on so(4) <cit.>, as well as systems invariant w.r.t. rotations <cit.>, <cit.>,  see also examples discussed in <cit.>, <cit.>. Unlike non-degenerate singularities, however, in the literature on topology and singularities of integrable systems there are only few papers devoted to degenerate singularites including parabolic ones.  We refer, first of all, to  the following six —  L. Lerman, Ya. Umanskii <cit.>,  V. Kalashnikov <cit.>, N. T. Zung <cit.>, H. Dullin, A. Ivanov <cit.>,  K. Efstathiou, A. Giacobbe <cit.> and Y. Colin de Verdière <cit.> — which we consider to be very important in the context of general classification programme for bifurcations occurring in integrable systems.  

It is well known that from the smooth point of view,
all parabolic orbits are equivalent,  i.e. any two parabolic orbits admit fiberwise diffeomorphic neighborhoods   (Lerman-Umanskii <cit.>, Kalashnikov <cit.>).  The same is true for cuspidal tori  <cit.>.  The simplest model for a parabolic singularity is as follows.

Consider the direct product of ^3 with coordinates x,y,λ and a circle  S^1  parametrized by φ 2π  and two functions on this product ^3 × S^1:

    H = x^2 + y^3 + λ y      F = λ.

They commute with respect to the symplectic form

    Ω = dx∧ dy + dλ∧ dφ.



The curve γ_0(t)=(0, 0, 0, t) is a parabolic orbit of an integrable Hamiltonian system defined by commuting functions H and F. However, in general,  we cannot assume that these coordinates x,y,λ, φ are canonical  (in other words, the formula for Ω could be different).  




The starting point of the present paper was the following question. We know that elliptic and hyperbolic orbits have no symplectic invariants <cit.>. In other words,  for any elliptic or hyperbolic  (with orientable or non-orientable separatrix diagram) orbit there exists a symplectic canonical form, one and the same for all orbits of a given type (see, e.g., <cit.>). Is the same true for parabolic orbits or they admit non-trivial symplectic invariants?

It appears that non-trivial symplectic invariants do exist  (a very simple invariant is given by Proposition <ref>).  Moreover, we show that all symplectic invariants of parabolic orbits can be expressed in terms of action variables (Theorem <ref>).   The next natural step would be to extend a fiberwise symplectomorphism between tubular neighborhoods of two parabolic orbits to saturated neighborhoods of the cuspidal tori that contain these orbits. This is done in Section <ref>:   Theorems <ref> and <ref> (see also Remark <ref>) give necessary and sufficient conditions for symplectic equivalence of cuspidal tori.  The latter theorems basically say the only symplectic semi-local invariant of a cuspidal torus is the canonical integer affine structure on the base of the corresponding singular Lagrangian fibration.  In other words, cuspidal tori satisfy the following principle formulated  in   <cit.>:

Let ϕ : M → B and ϕ': M' → B' be two singular Lagrangian fibrations. If B and B' are affinely equivalent (as stratified
manifolds with singular integer affine structures), then these Lagrangian fibrations are fiberwise symplectomorphic.

Also we would like to notice that although parabolic singularities are rather simple and specific,  some techniques developed and used in this paper are quite general and can be used for analysis of more complicated singularities.  They also can be generalised to the case of many degrees of freedom.  


Acknowledgements:    This work was supported by the Russian Science Foundation (project no. 17-11-01303).   We are very grateful to D. Guzzetti,  A. Varchenko, S. Nemirovskii, N. T. Zung, A. Izosimov  for valuable comments and discussions. 









§ DEFINITION OF PARABOLIC SINGULARITIES.  CANONICAL FORM WITH NO SYMPLECTIC STRUCTURE
 



    
We begin with the definition of parabolic orbits following <cit.> [One essential condition, in our opinion, is missing in  <cit.>.  In  Definition <ref> below we make a necessary modification by adding Condition (iii).]. Let H and F be a pair of Poisson commuting real-analytic functions on a  real-analytic symplectic manifold (M^4, Ω).  They define a Hamiltonian ^2-action  (perhaps local) on M^4.  The dimension of the ^2-orbit through a point P∈ M^4 coincides with the rank of  the differential of the momentum map ℱ=(H,F): M^4 →^2 at this point and we are interested in one-dimensional orbits.  Without loss of generality throughout the paper we will assume that dF(P) 0.   Consider the restriction of H onto the three-dimensional level set of F through P, that is,  H_0:=H|_{F=F(P)}.  We assume that the rank of dℱ at the point P equals one.  This is equivalent to any of the following:



  * P is a critical point of H_0;


  * there exists a unique k∈ such that dH(P) = k dF(P), in particular, P is a critical point of F-kH.


These properties hold true for each singular point P of rank one of the momentum mapping ℱ=(H,F) under the condition that dF(P) 0.


 
A point P (and the corresponding ^2-orbit through this point) is called parabolic if the following conditions hold:


  (i) the quadratic differential d^2H_0(P) has rank 1; 

  (ii) there exists a vector v ∈ d^2H_0(P) such that v^3H_0 0 (by v^3H_0 we mean the third derivative of H_0 along the tangent vector v at P);

  (iii) the quadratic differential d^2(H-kF)(P) has rank 3, where k is the real number determined by the condition dH(P)=kdF(P).








In this definition, we use the third derivative of a function along a tangent vector which, in general,  is not well defined.  In our special case, however,  this derivative makes sense as dH_0(P)=0 and v∈ d^2H_0(P).  These two properties allows us to define it as follows:

    v^3 (H_0) = d^3/dt^3 |_t=0 H_0 (γ (t)),

where γ (t) is an arbitrary  curve on the hypersurface { F = F(P)} such that  γ(0)=P, dγ/dt (0) = v. The result does not depend on the choice of γ(t). Indeed,  

    d^3/dt^3 H_0(γ(t)) =   d^3H_0 (γ', γ',γ') + 3d^2H_0(γ',γ”) + dH_0(γ”') = d^3H_0 (γ', γ',γ') = d^3H_0 (v,v,v),
 
as dH_0=0 and γ' ∈ d^2H_0. This computation also shows that the third differential d^3H_0 is a well-defined cubic form on d^2H_0 so that condition (ii) is equivalent to the fact that the third differential d^3H_0 does not vanish on d^2 H_0  (at the point P). 




It can be checked that in  Definition <ref>, we may replace H and F by any other independent functions H =  H (H, F),  F =  F(H,F) such that d F(P)  0. In other words, the property of being parabolic  refers to a singularity of  the momentum mapping ℱ:  M^4 →^2 and does not depend on the choice of local coordinates in a neighborhood of ℱ(P) ∈^2.  Necessary details can be found in Appendix, see Proposition <ref>.



The following statement describes the structure of the singular Lagrangian fibration in a neighborhood of a parabolic point P.  As we are mostly interested in this fibration (rather than specific commuting functions H and F), we allow ourselves to replace H with H = H(H,F) where ∂H/∂ H 0 and to shift  and change the sign of F, so that H and F= ± F+const  still commute and define the same Lagrangian fibration as H and F.  Notice that according to Remark <ref>,  P is parabolic for H and F. 


 
Locally in a neighborhood of P  there exist a transformation 

    H   = H(H,F),    ∂H/∂ H 0, 
    F   = ± F + const,

and a local coordinate system x,y,λ, φ such that  (x,y,λ,φ)|_P=(0,0,0,0) and

    H = H(H,F)= x^2 + y^3 + λ y     F = λ.




We do not require that this coordinate system is canonical and, in this view,  Proposition <ref> describes a normal form of a parabolic singularity in the sense of Singularity Theory with no symplectic structure involved.  This statement is local and we do not need to assume that the orbit through P is closed. Later on, the variable φ will be one of the angle variables defined modulo 2π,  but here φ just  belongs to a certain interval.



   
The proof of this statement if well known but we still want to briefly explain some of its steps to reveal  important underlying phenomena. The first step is to find x,y,λ,φ without touching H and F. 

 
Under the above assumptions,  there exist local coordinates x,y,λ,φ such that (x,y,λ,φ)|_P=(0,0,F(P),0) and

    H = ± (x^2  + y^3 + b(λ) y + a(λ)),        F=λ,

where a(λ) and b(λ) are real-analytic functions with b(F(P))=0, b'(F(P))0.



 
Without loss of generality, we assume that H(P)=F(P)=0.
First of all we need to kill one dimension using the fact that H and F Poisson commute.  Since  dF(P) 0 we can choose a canonical coordinate system p_1, q_1, p_2, q_2 such that F=q_2.  Since H and F commute,  we conclude that H does not depend on p_2, i.e., H=H(p_1, q_1, q_2). Thus, p_2 does not play any role, so we may forget about it and continue working with p_1, q_1, q_2.


Let us now think of H as a function of two variables q_1 and p_1 depending on q_2 = λ as a parameter.
We have ∂ H/∂ p_1|_P=∂ H/∂ q_1|_P=0 and, without loss of generality, ∂^2H/∂ p_1^2|_P0.
We are now in a quite standard situation in singularity theory.


By a parametric version of the Morse lemma, the function H can be written as H=±(x^2+f(q_1,λ)), for some new local variable x=x(p_1,q_1,λ) such that x|_P=0 and x/ p_10. 
Now, condition (ii) of the definition of a parabolic point is satisfied if and only if the function f(q_1,0) in one variable q_1 has order 3 at the point q_1|_P. Hence, this function can be written as ŷ^3 for some variable ŷ=ŷ(q_1) with ŷ(q_1(P))=0.


Now the function f(q_1,λ) is a 1-parameter “deformation” of the function f(q_1,0)=ŷ^3 with the parameter λ. It follows from <cit.> that the deformation ŷ^3+λ_2ŷ+λ_1 is right-infinitesimally versal. By the versality theorem <cit.>, it is right-versal (for a definition of a versal deformation, see <cit.>). Since any deformation is right-equivalent to a deformation induced from the right-versal one, we have f(q_1,λ)=y^3+b(λ)y+a(λ) for some real-analytic functions y=y(ŷ,λ), a(λ) and b(λ) such that y(ŷ,0)=ŷ, a(0)=b(0)=0. Since, by assumption, the quadratic differential d^2(H-kF)(P) has rank 3, we have b'(0)0.
So, we obtain the representation (<ref>).
 



 
Later on we will need to rearrange leaves of our singular Lagrangian fibration by using some transformations of the form  (the fibration itself remains unchanged) 

    H ↦H=H(H,F),    F ↦F=F(H,F).

So we need to understand if such a transformation (acting on the base of the Lagrangian fibration) can be realised by a fiberwise analytic diffeomorphism upstairs.  In other words, we want to know which of transformations (<ref>) are liftable.

Let us look at the (local) bifurcation diagram (i.e. the set of critical values) of the map defined by H and F from (<ref>). This bifurcation diagram is as follows, for a + sign in (<ref>):

    Σ = {(H-a(F))^2 = -4/27 b(F)^3 }⊂^2 (H,F),

and it has a cusp at the point (H(P),F(P)) that splits Σ into two smooth branches, Σ_ell and Σ_hyp, corresponding to one-parameter families of elliptic and hyperbolic orbits. The bifurcation diagram for a(λ)=0 and b(λ)=λ is shown on Fig. 3. 
Notice that our choice of the + sign in (<ref>) simply means that the monotone function H-kF|_Σ increases w.r.t. the orientation of Σ from Σ_ell to Σ_hyp.

It can be easily seen (see the proof of Proposition <ref> below) that this bifurcation diagram allows us to reconstruct  both functions a(λ) and b(λ).    We will use this observation to prove the following


Assume we have two parabolic singularities defined by functions H, F at a point P and H, F at a point P respectively.  A map (local  analytic diffeomorphism)

    ϕ :  ^2 (H, F) →^2 (H, F)

is liftable if and only if ϕ transforms the bifurcation diagram  of (H,F) to that of (H, F), i.e. ϕ (Σ) = Σ, together with its partition into elliptic and hyperbolic branches. In other words, the condition ϕ (Σ) = Σ is necessary and sufficient for the existence of a local  analytic diffeomorphism Φ such that the diagram 

    M^4 r[]Φd(H,F)   M^4  d[](H,F)
    ^2 rϕ   ^2

is commutative.
 



The “only if” part is obvious. 

Let us prove the “if” part.
Denote ϕ∘(H,F) by (H_1,F_1). Clearly, ϕ transforms the bifurcation diagram of (H,F) to that of (H_1,F_1), together with their partitions into elliptic and hyperbolic branches. Hence, the bifurcation diagram Σ_1 of (H_1,F_1) coincides with the bifurcation diagram Σ of (H,F), together with its partition into elliptic and hyperbolic branches.


As shown above,  under the condition that dF( P)0,  the bifurcation diagram Σ of the mapping ℱ=( H,  F): M^4 →^2(h,f) is defined by

    Σ ={ (h,f)∈^2  |    (h- ã(f))^2 = -4/27b̃(f)^3}

for some functions a(·) and b(·) determined by the canonical form (<ref>).
Hence Σ lies entirely in a half-plane {(h,f) |b(f)≤ 0}⊂^2(h,f)  bounded by a line {f=} through the cusp point (H(P),F(P)).  Since Σ_1=Σ, we conclude that dF_1(P)0 as well. 
 
By Lemma <ref>, there exist local (real-analytic) coordinates x_1,y_1,λ_1,φ_1 in a neighborhood U_1 of P and coordinates x,y,λ,φ in a neighborhood U of P such that 

    η_1 H_1    = x_1^2  + y_1^3 + b_1(λ_1) y_1 + a_1(λ_1),          F_1    = λ_1, 
    ηH   = x^2  + y^3 + b(λ) y + a(λ),         F   = λ,

for some signs η_1,η∈{1,-1}. 
The elliptic and hyperbolic branches of Σ have the form

    Σ_ell   = {. (h,f)= (η(a(f) - 2(-b(f)/3)^3/2),f)  | b(f)<0}, 
    Σ_hyp   = {. (h,f)= (η(a(f) + 2(-b(f)/3)^3/2),f)  | b(f)<0},

in particular, 
ηh-a(f)>0 on the hyperbolic branch of Σ and <0 on its elliptic branch.
Since similar properties and formulae hold for the elliptic and hyperbolic branches Σ_1,ell and Σ_1,hyp of Σ_1, 
moreover Σ_1,ell=Σ_ell and Σ_1,hyp=Σ_hyp,  we obtain the equalities

    η_1=η,   
    a_1(λ)=a(λ),   
    b_1(λ)=b(λ)

where the equalities of functions hold in a half-neighbourhood {λ|b(λ)≤0} of the point F(P)∈. Since all functions are real-analytic at this point, the equalities of functions in (<ref>) hold in an entire neighbourhood.

Define a (real-analytic) diffeomorphism germ Φ:(U_1,P)→(U,P) given by the identity map in the local coordinates (x_1,y_1,λ_1,φ_1) and (x,y,λ,φ). By (<ref>) and (<ref>), Φ transforms (H,F) to (H_1,F_1), so it 
has the desired property ϕ∘(H,F)=(H_1,F_1)=(H,F)∘Φ.






Proposition <ref>  implies the following



Let P be a parabolic point for an integrable Hamiltonian system with the momentum mapping ℱ= (H,F): M^4 →^2. Assume that the local bifurcation diagram Σ⊂^2(H,F) of ℱ takes the standard form 

    Σ = {H^2 = -4/27F^3 }    Σ_ell = Σ∩{H<0}, Σ_hyp = Σ∩{H>0}.

Then in a neighborhood of a parabolic point  there exists a local coordinate system (x,y,λ,φ)  in which 
H=x^2 + y^3 + λ y and F=λ.



   
It is sufficient to notice that the pair of functions H = x^2 + y^3 + λ y, F = λ define a parabolic singular point with the standard bifurcation diagram (<ref>).   According to Proposition <ref>  any other parabolic singularity with the same bifurcation diagram is fiberwise diffeomorphic to this simplest model, moreover,  the map ϕ: ^2(H,F) →^2( H,  F) between the bases is defined by H = H, F=F.  


We are now able to complete the proof of Proposition <ref>. In view of Corollary <ref>, it is sufficient to show that by a suitable transformation (<ref>) the bifurcation diagram, together with its partition into elliptic and hyperbolic branches, can be reduced to the standard form (<ref>). 

As shown above, for the original functions H and F the bifurcation diagram is defined by the equation 

    Σ = {(H-a(F))^2 = -4/27 b(F)^3 }

(here we assume that H in (<ref>) comes with +).

Let  F(P) = f_0 so that b(f_0)=0 and b'(f_0)0,  then we can represent b(λ) as b(λ) = (λ-f_0) c(λ) with c(f_0) 0  and rewrite the equation for Σ in the form

    ( H -a(F)/|c(F)|^3/2)^2 = -η_F 4/27 (F-f_0)^3

with η_F=c(f_0)/|c(f_0)| or, equivalently, 

    Σ = {H^2 = -4/27F^3 }    Σ_ell = Σ∩{H<0}, Σ_hyp = Σ∩{H>0},

for H = H-a(F)|c(F)|^3/2 and F = η_F (F - f_0), which coincides with (<ref>) as required.  















§ DESCRIPTION OF A NEIGHBORHOOD OF A PARABOLIC ORBIT WITH  SYMPLECTIC STRUCTURE
 




Our next goal is to describe the symplectic structure Ω near a parabolic orbit. 


An important property of a parabolic orbit is the existence (in real-analytic case) of a free Hamiltonian S^1-action in its tubular neighborhood  (N.T. Zung  <cit.>, compare Kalashnikov <cit.>).
In other words, without loss of generality we may assume that one of the commuting functions, say F, generates this S^1-action, i.e., the Hamiltonian flow of F is 2π-periodic. 
From the viewpoint of singularity theory, this means that in our case the parameter of the versal deformation is essentially unique and is given by the Hamiltonian of the S^1-action  (or in slightly different terms, by the action variable related to the cycle in the first homology group of fibers that corresponds to this S^1-action).  The latter interpretation, in particular, means that one of two action variables is a real-analytic function defined on the whole neighborhood U(ℒ_0) of ℒ_0 including singular fibers, where ℒ_0 denotes the singular fiber (cuspidal torus) containing the parabolic orbit γ_0.
The action variable F is defined up to changing F→± F+, and we can (and will) choose F in such a way that F(P)=0 and the bifurcation diagram Σ is located in the domain { F ≤ 0}. 


Basically, what we want to do next is to reduce our Hamiltonian system w.r.t. this action.  We shall think of F as a parameter and denote it by λ  as above.
In particular, now we can choose a 
coordinate system x,y,λ,φ in a tubular neighborhood U(γ_0) of γ_0 in such a way that the Hamiltonian vector field of λ is ∂/∂φ. Since H commutes with F=λ, we conclude that H=H(x,y,λ) and we are in the situation discussed in the previous section. If we are only interested in the symplectic topology of the fibration, we are free in the choice of H (in contrast to F which is essentially unique), so according to Proposition <ref> we may assume  without loss of generality 
that H= x^2 + y^3 + λ y. However, these coordinates are not canonical, so that (in the tubular neighborhood U(γ_0)) the symplectic structure takes the following form (here we take into account the condition that Ω is closed and the Hamiltonian vector field of λ is ∂/∂φ  or, equivalently,    i_/φΩ = -dλ):  

    Ω   = f(x,y,λ) dx∧ dy + dλ∧ dφ +  dλ∧(P(x,y,λ) dx + Q(x,y,λ) dy)=
       =ω_λ + dλ∧ dφ + .




The form ω_λ = f(x,y,λ)dx∧ dy  can be considered as the restriction of Ω onto the common level of λ and φ (we assume that φ=0 but λ  varies and is considered as a parameter).   The other interpretation of ω_λ is that it is the one-parameter family of symplectic forms obtained from Ω by the reduction w.r.t. the Hamiltonian S^1-action  (or, using old-style terminology, w.r.t. the cyclic variable φ). Here is a more formal statement.


 
In a tubular neighborhood of a parabolic orbit γ_0 we can choose a coordinate system x,y,λ,φ (with  φ mod 2π∈ / 2πℤ)  such that (x,y,λ)|_γ_0 = (0,0,0) and our singular Lagrangian fibration is given by two functions

    F=λ     H=x^2 + y^3 + λ y

and the symplectic form 

    Ω = f(x,y,λ) dx∧ dy   +   dλ∧ dφ  +

as in (<ref>).  □




Without loss of generality we may assume that f(x,y,λ)>0    in (<ref>). Indeed,  in order for the latter property to be fulfilled, we only need to replace x with -x if necessary.  We also notice that since Ω is closed,  formula (<ref>) can be rewritten as

    Ω =  dX(x,y,λ) ∧ dy + dλ∧ dφ

for a certain real-analytic function X(x,y,λ) with X/ x>0 and φ = φ + R(x,y, λ) for some real-analytic function R(x,y,λ).



It follows from Proposition <ref> that the function F is uniquely defined (being a generator of the S^1-action), but H is not.  However H cannot be chosen arbitrarily because the bifurcation diagram for F and H  must be of a very special form, namely (<ref>).  If this condition is fulfilled then H is allowed and, using Corollary <ref>,   we can modify Proposition <ref>  in the following way.  


Consider a tubular neighborhood of a parabolic trajectory.  Let H and F be two functions defining our fibration and satisfying the following conditions:


  (i) the bifurcation diagram of (H,F) is canonical, i.e., as in  (<ref>);

  (ii) F is  2π-periodic, i.e., is a generator of a free Hamiltonian S^1-action. 

Then there exists a coordinate system (x,y,λ,φ)  as in Proposition <ref>. □




It follows from Proposition <ref> that if we are given two integrable systems with parabolic trajectories, we can always find a fiberwise real-analytic diffeomorphism between their tubular neighborhoods that respects the S^1-actions and corresponding periodic Hamiltonians.  This means that without loss of generality we may assume that we are given just one single fibration defined by H and F having canonical form (<ref>)  with two different symplectic forms given by (<ref>)  (i.e. such that H and F commute and the Hamiltonian vector field of λ is ∂/∂φ):

    Ω = ω_λ + dλ∧ dφ +

and

    Ω=ω̃_λ + dλ∧ dφ + .



We still have two different integrable systems but after the above “pre-identification” they have many common properties. Namely, 




  (i) They have a common local coordinate system (x,y,λ,φ) from Proposition <ref>;

  (ii) F=λ is a 2π-periodic integral for the both systems;

  (iii) The S^1-actions defined by F for Ω and Ω  coincide  (i.e.,  X_F =  X_F=∂/∂φ where X_F and X_F denote the  Hamiltonian vector fields generated by F w.r.t. Ω and Ω  respectively);

  (iv) The bifurcation diagrams of these two systems coincide;

  (v) The orientations and co-orientations of the parabolic trajectory  γ_0(t)=(0,0,0,φ=t) induced by Ω and Ω coincide  (see Section <ref>, Theorem <ref>). 



We need to find out whether Ω can be transformed to Ω by a suitable fiberwise diffeomorphism Φ. First, we  impose a stronger condition on Φ by requiring that Φ preserves not only the fibration but also each particular fiber, i.e. the functions H and F  (in other words, rearrangements of fibers are temporarily forbidden, i.e.  Φ induces the identity map on the base of the fibration). 



The following statement reduces this 4-dim problem for Ω and Ω to a similar problem for the reduced forms ω_λ and ω_λ (in other words, we now reduce our “two-degrees-of-freedom” problem to a parametric “one-degree-of-freedom” problem).


Consider the singular fibration defined by the functions H=x^2 + y^3 + λ y and F=λ.  This fibration is obviously Lagrangian w.r.t. any of the symplectic structures (<ref>) and (<ref>) in a neighborhood of the parabolic orbit γ_0={x=y=λ=0}.


 The following two statements are equivalent.


  (i) In a tubular neighborhood of the parabolic orbit γ_0 there is a (real-analytic) diffeomorphism Φ  such that


  * Φ preserves H and F;

  * Φ^* (Ω) = Ω.


  (ii) There exists a one-parameter family of local diffeomorphisms ψ_λ (x,y) (real-analytic in x, y and λ) leaving fixed the origin in ^2(x,y) at λ = 0 and such that, for each λ∈ close enough to 0,


  * ψ_λ preserves H(x,y,λ);

  * ψ_λ^* (ω_λ) = ω_λ.


 

Roughly speaking, this statement says that the additional terms in (<ref>) and (<ref>) are not important and can be ignored.
We also remark that we can replace the conditions that H and F are preserved by saying that the fibration is preserved.



 

The fact that (i) implies (ii) is almost obvious.  Indeed, since Ω and Ω are of quite special form, Φ^*(Ω)=Ω and F=λ is preserved, then in local coordinates x,y,λ,φ, the diffeomorphism Φ takes the following form:



    x̃   = x̃ (x,y,λ), 
    ỹ   = ỹ (x,y,λ),
    λ̃   = λ, 
    φ̃   = φ + R(x,y,λ),

then if we consider the first two functions as a family of diffeomorphisms ψ_λ (x,y), then we will immediately see that (ii) holds.
Since Φ preserves H and F, it leaves invariant the set of such points (x,y,λ,φ) that dH(x,y,λ,φ) and dF(x,y,λ,φ) are proportional. But for λ = 0 this set coincides with γ_0, so Φ maps γ_0 to itself. Therefore ψ_0(0,0)=(0,0).



The proof of the converse statement consists of two steps. Assuming that ψ_λ (x,y) satisfies the conditions from (ii), we define Φ_1 as follows:

    (x̃, ỹ )    = ψ_λ (x,y), 
    λ̃   = λ, 
    φ̃   = φ.

It is easily checked that, for this Φ_1, the symplectic forms Φ_1^*(Ω) and Ω coincide up to additional terms, that is

    Φ_1^* (Ω) - Ω = dλ∧(P(x,y,λ) dx + Q(x,y,λ) dy).

Hence, our goal is to show that these additional terms do not play any essential role and can be killed by an appropriate shift φ↦φ - R(x,y,λ) (without changing the other coordinates). In other words, we need to find R(x,y,λ) such that dλ∧ dR(x,y,λ) = dλ∧(P(x,y,λ) dx + Q(x,y,λ) dy). The existence of such a function follows immediately from the closedness of the form  
dλ∧(P(x,y,λ) dx + Q(x,y,λ) dy)  (this form is the difference of two closed forms Φ_1^* (Ω) and Ω. Finally, we define Φ as the composition of Φ_1 and the above shift, 
and we get Φ^* (Ω) = Φ_1^* (Ω) - dλ∧ dR(x,y,λ) =Ω due to (<ref>).

It remains to notice that, since ψ_0(0,0)=(0,0) and γ_0={x=y=λ=0}, we have Φ(γ_0)=γ_0, thus Φ is defined in a neighborhood of γ_0 as required.



Our next observation is that symplectic invariants do exist, in other words,   the desired map Φ (or, equivalently, the family ψ_λ) may not exist.  Moreover, the existence of just one map ψ_0 implies rather strong condition.  To show this, we treat the case λ = 0 in detail.















§ THE CASE Λ=0,  ONE-DEGREE OF FREEDOM PROBLEM
 




In this Section, for notational convenience, we use a different sign in the definition of H.  Consider the function H= y^3-x^2  (in a neighborhood of the origin) and two symplectic forms ω_0 and ω_0 (all of our objects are real-analytic).   We want to know necessary and sufficient conditions for the existence of a local diffeomorphism ψ_0   satisfying ψ_0^* ω_0 = ω_0 and   (two versions):


  * either preserving H  (strong condition);

  * or preserving the (singular) fibration defined by H (weaker condition) (more formally,   ψ_0^* (H) = h(H)  where h(H) is real-analytic and h'(0) 0).





The complex version of the first problem was studied in <cit.>, in this Section we adapt some of these results to the real case we are considering. In the following, ℝ{H} and ℂ{H} will denote, respectively, real-analytic germs and complex-analytic germs in the variable H at 0, i.e., convergent power series in the respective fields. Consider H = y^3-x^2 as a holomorphic function, we have:

 
Any holomorphic 2-form ω_0  can be decomposed, in a sufficiently small neighborhood U of 0 ∈ℂ^2, as follows

    ω_0 = α(H) dx ∧ dy + β(H) y dx ∧ dy + d H ∧ d η

for some holomorphic germ η(x,y), and unique α, β∈ℂ{H}.


 
If ω_0 is symplectic, then α(0) ≠ 0.


In our case we are dealing with real objects ω_0 and H, in this case  α(H), β(H) are  real-analytic, and η(x,y) can be chosen to be real-analytic, as can be shown by taking the real part of Equation (<ref>).

Choose two one-dimensional cross-sections N_1, N_2 to the fibration defined by H as shown in Fig.1.  Each non-singular leaf τ_H of this fibration (with a given value of H) now will be interpreted as a trajectory of the Hamiltonian vector field X_H = ω_0^-1(dH)  with respect to the symplectic form ω_0.  For each trajectory τ_H we can measure the passage time  Π(H) from N_1 to N_2.  This function can be expressed as

    Π(H) = ∫_N_1^N_2ω_0/d H

(integral taken along the trajectory τ_H) where ω_0/dH is the  Gelfand-Leray form associated to the pair (ω_0,H), i.e., any 1-form γ defined in the region dH ≠ 0 and such that dH∧γ= ω_0 (the form γ is not uniquely defined, but its restriction to the level-sets H= const is unique).

We can similarly consider the area function 𝖺𝗋𝖾𝖺(H) defined as the integral of  ω_0 over the subset  
of {0 ≤ H(x,y) ≤ H} bounded by the sections N_1, N_2.  As a consequence of Fubini's theorem,
one has

    d 𝖺𝗋𝖾𝖺(H)/d H =Π(H).


Clearly,  Π(H) is a real-analytic function defined for all  (small) H.   As H tends to 0, the passage time Π(H) tends to infinity and it is natural to look at the asymptotic behaviour of Π(H) at zero. 




The function Π(H) for H>0 can be written as

    Π(H) = a(H) H^-1/6 + b(H) H^1/6 + c(H),     H>0,

where a, b, c ∈ℝ{H}. 

Moreover,
a (H)= C_0 α(H) and b(H) = C_1 β(H) for some non-zero constants C_0, C_1 ∈ℝ, with C_0> 0 and C_1< 0.



Before proving the lemma, we give some remarks:




  * The functions in this representations are uniquely defined, i.e., if

    a(H) H^-1/6 + b(H) H^1/6 + c(H) = ã(H) H^-1/6 + b̃(H) H^1/6 + c̃(H),

then a(H)=ã(H),  b(H)=b̃(H) and c(H)=c̃(H).


  * If we change the sections N_1 and N_2 by a deformation in the class of such sections, then the function Π(H) changes by adding a certain analytic function, given by the passage time between the old and the new sections. However, if we replace N_1 and N_2 by each other, then the function Π(H) will be replaced by -Π(H). This shows that the functions a(H) and b(H) (up to multiplying with -1 simultaneously) do not depend on the choice of the cross-sections N_1 and N_2. Since we are working with a symplectic form, we have α(0)0 and a(0) ≠ 0, and we can be more specific: the functions a(H) and b(H) with a(0)>0 do not depend on the choice of the cross-sections N_1 and N_2.


  * In a similar way,  we can define the functions ã, b̃ and c̃ for the second symplectic structure ω_0.   If ψ preserves H and transforms ω_0 to ω_0,   then the Hamiltonian vector field X_H will be transformed to the Hamiltonian vector field X_H = ω_0^-1(dH)  (with the same Hamiltonian H).   Since ψ does not preserve the cross-sections N_1 and N_2, the passage time Π(H)  will, in general, differ from Π (H) by adding some analytic functions (and, possibly, by multiplying with -1),  which shows that the functions a(H) and b(H)  with a(0)>0 remain invariant under ψ, i.e. a(H)=ã(H),  b(H)=b̃(H), provided that ã(0)>0 too. In other words, a(H) and b(H) with a(0)>0 are symplectic invariants  (under the condition that ψ preserves H).


  * It is easy to give an example of two symplectic structures producing two different pairs of functions a and b in the asymptotic decomposition (<ref>).





 
Consider the decomposition (<ref>). Taking the integral of the Gelfand-Leray form we get:

    Π(H) = α(H)∫_N_1^N_2dx ∧ dy/d H + β(H) ∫_N_1^N_2y dx ∧ dy/dH +  N_2^*η(H)-N_1^*η(H)

(the coefficients can be taken outside of the integral, since we integrate along a trajectory τ_H where H is constant). The last two terms give a real-analytic contribution. To finish the proof it is sufficient to show that, for H>0

    ∫_N_1^N_2dx ∧ dy/d H - C_0 H^-1/6∈ℝ{H},     ∫_N_1^N_2y dx ∧ dy/d H -  C_1 H^1/6∈ℝ{H},

for some non-zero real constants C_0, C_1, so  that a(H) = C_0α(H) and b(H)= C_1β(H). We can assume that N_1 = {x = 1} and N_2 = {x = -1}. We have

    y^j dx∧ dy/dH = -dx/3 y^2-j,   j = 0,1.

Hence, we are reduced to compute, for j = 0,1, the integral:

    J_j(H)   =-1/3∫_1^-1y(H,x)^j-2 dx = 2/3∫_0^1(H+x^2)^j-2/3 dx
       =2/3 H^j-2/3∫_0^1(1+x^2H)^j-2/3 dx 
    =1/3H^j-2/3∫_0^1t^12-1(1+tH)^j-2/3 dt
       =2/3H^j-2/3F(2-j3,12,32; -1H)

where F(p,q,r;z) is the hypergeometric function. In this case we can use the connection formula (<cit.>)

    F(p,q,r;z)    =    c_1(-z)^-pF(p, 1+p-r,1+p-q; 1/z)+
         +c_2(-z)^-qF(q, 1+q-r,1+q-p; 1/z)

where 

    c_1 = Γ(r)Γ(q-p)/Γ(r-p)Γ(q),    
    c_2 = Γ(r)Γ(p-q)/Γ(r-q)Γ(p).


This gives:

    J_j(H)    = 23H^j-2/3(c_1H^2-j/3F(2-j3,1-2j6,7-2j6;-H)+c_2H^1/2F(12,0,5+2j6;-H))
       = 23c_1 F(2-j3,1-2j6,7-2j6;-H) + 23c_2H^2j-1/6
       = C_j H^2j-1/6 + d_j(H),      d_j∈ℝ{H},

where C_0 = √(π)/3Γ(1/6)/Γ(2/3) and C_1 = √(π)/3Γ(-1/6)/Γ(1/3). This proves (<ref>) as required.



For r ∈ℚ, consider the operator ϕ_r: ℝ{H}→ℝ{H} defined by ϕ_r: A(H) ↦ A'(H) H + rA(H). If r ∉ℤ then ϕ_r is bijective.



The function 𝖺𝗋𝖾𝖺(H) for H≥0 can be written as

    𝖺𝗋𝖾𝖺(H) = A(H) H^5/6 + B(H) H^7/6 +  C(H),     H ≥ 0,

where A, B, C∈ℝ{H} are the unique real-analytic germs such that

    a(H) = A'(H) H + 56A(H),   b(H) =B'(H) H + 76B(H),   c(H)=C'(H),  C(0)=0,

in other words A= ϕ_5/6^-1(a), B = ϕ_7/6^-1(b).  □





Let ω_0,ω_0 be two real-analytic symplectic forms. Suppose that ω_0 - ω_0 = dH ∧ dη for some real-analytic function germ η(x,y) at 0∈ℝ^2. Then there is a local diffeomorphism ψ at 0 ∈ℝ^2 such that ψ^*H = H and ψ^*ω_0= ω_0.


 
We adapt the proof of <cit.>. Put ω_t = ω_0 + t(ω_0-ω_0). In a neighborhood of zero, the forms  ω_t, for t ∈ [0,1], are also non-degenerate, indeed the equation ω_0 - ω_0 = dH ∧ dη implies that the two forms have the same sign/orientation at zero and near zero. 
Since ω_t is a convex combination of functions with the same sign, it will also be non-zero in a neighborhood of zero. Define a real-analytic time-dependent vector field X_t by

    i_X_tω_t = -η d H.

Let ϕ_t be the flow generated by such X_t, we can integrate it for t ∈ [0,1]. Notice that

    L_X_tω_t = i_X_tdω_t  + d i_X_tω_t = d H∧ dη = ω_0-ω_0,

therefore

    d/d tϕ_t^*ω_t = ϕ^*_t(L_X_tω_t + d/d tω_t) = 0,

so that ϕ_1^*ω_0 = ω_0. Moreover L_X_tH =0, because of the equality:

    0 = i_X_t(d H∧ω_t) = (i_X_td H)ω_t+ d H∧ i_X_tω_t = (L_X_tH)ω_t+d H∧ i_X_tω_t= (L_X_tH)ω_t,

and using  ω_t ≠ 0. This means that H∘ϕ_t = H. Finally take ψ = ϕ_1.



In the following we will specify as a subscript the symplectic structure ω_0 in the notation for α,β, a,b and A,B, that is, writing α_ω_0,β_ω_0, a_ω_0,b_ω_0 and A_ω_0,B_ω_0. In the rest of the Section, for the reasons explained in the second and third remarks below Lemma <ref>, we will consider symplectic forms inducing a fixed orientation. In this regard we can consider, without loss of generality, only  symplectic forms ω satisfying  α_ω(0) >0.  Such a symplectic form is said to be positively-oriented.


 In the above setting and notation we come to the following statement:



Let ω_0, ω_0 be positively-oriented symplectic forms. An H-preserving map ψ  such that ψ^*ω_0=ω_0 exists, if and only if the following conditions hold:

    α_ω_0(H) =  α_ω_0(H) β_ω_0(H)= β_ω_0(H)

or, equivalently, a_ω_0(H) = a_ω_0(H)  b_ω_0(H)= b_ω_0(H) or   A_ω_0(H) =  A_ω_0(H)  B_ω_0(H)= B_ω_0(H).


 Sufficiency: suppose α_ω_0 = α_ω_0 and β_ω_0 = β_ω_0, then ω_0 - ω_0 = dH ∧ dη for some real-analytic germ η, and Theorem <ref> proves the assertion. From Lemma <ref> and Corollary <ref> we know that equalities of any of these invariants  are equivalent.

Necessity: suppose ψ exists, let us prove that the invariants coincide. Since ψ preserves H and sends ω_0 to ω_0, we conclude (due to Lemma <ref> and the third remark below it) that a_ω_0(H) = a_ω_0(H) and b_ω_0(H) =  b_ω_0(H), which implies α_ω_0(H) = α_ω_0(H) and β_ω_0(H) = β_ω_0(H).


It follows from the above proposition, together with the first remark below Lemma <ref>, that:


Let ω_0, ω_0 be positively-oriented symplectic forms.
An H-preserving map ψ  such that ψ^*ω_0=ω_0 exists if and only if Π_ω_0(H) - Π_ω_0(H), which is defined on {H>0}, extends to a real-analytic function in a neighborhood of H=0.□


We can also reformulate this result in terms of normal forms.


For H= y^3-x^2 and ω_0=f(x,y) dx∧ dy there is a real-analytic local coordinate system u, v and germs α,β∈ℝ{H} such that 

    H=v^3 -u^2    ω_0=  α(H) · du∧ dv + β(H) · v  du∧ dv.
  
For positively-oriented symplectic forms, the functions α(H) and β(H) are uniquely defined   (the coordinates u, v are not).□
 

 
Let us now see what happens if ψ does not preserve H, but transforms it to a function of the form h(H), h'(0) 0  (in fact h'(0)>0). Let ω_0, ω_0 be positively-oriented symplectic forms. We consider necessary and sufficient conditions for the existence of a local diffeomorphism ψ such that ψ^*ω_0 = ω_0 and ψ^*H = h ( H) with h'(0) >0,
i.e. a local symplectomorphism ψ making the following diagram commutative:

    (^2, 0) r[]ψdH     (^2, 0)    d[]H
    
    (, 0) rh    (, 0).



 
Suppose there exists ψ such that ψ^*ω_0 = ω_0 and ψ^*H = h(H) with h(H) = H· g(H), g(0)>0. Then we have the following relations:


  i)

    A_ω_0(H) = g(H)^5/6A_ω_0(h(H)),
     
    B_ω_0(H) = g(H)^7/6B_ω_0(h(H)),


  ii) 
    α_ω_0(H) =  g(H)^-1/6( g'(H) H + g(H)) α_ω_0(h(H)),
    β_ω_0(H) =  g(H)^1/6( g'(H) H + g(H))β_ω_0(h(H)),


  iii) 
    a_ω_0(H) =  g(H)^-1/6( g'(H) H + g(H)) a_ω_0(h(H)),
     
    b_ω_0(H) =  g(H)^1/6( g'(H) H + g(H))b_ω_0(h(H)).

 


For proving Lemma <ref>, we need the following:


For any real-analytic map h(H) with h(0)=0 and h'(0)>0 
there exists ψ  (local real-analytic diffeomorphism) such that 

    H(ψ (x,y)) = h (H(x,y)).

In other words, any local diffeomorphism germ H ↦ h(H) at 0 with h'(0)>0 is liftable.
 


 
Let h(H) = H· g(H). Define r_h(x,y):= ( g(H(x,y))^1/2x, g(H(x,y))^1/3y), then

    H(r_h(x,y)) = g(H(x,y)) H(x,y) = h(H(x,y)).






i) The integrals below are taken over subsets bounded by the sections N_1, N_2. For any H≥0, we have

    𝖺𝗋𝖾𝖺_ω_0(H)    = ∫_0 ≤ H(x,y) ≤ Hω_0 =  ∫_0 ≤ H(x,y) ≤ Hψ^*ω_0 
       = ∫_ψ(0 ≤ H(x,y) ≤ H)ω_0  + D(H) =  ∫_0 ≤ H(x',y') ≤ h(H)ω_0  + D(H) 
       = 𝖺𝗋𝖾𝖺_ω_0(h(H))+  D(H),      D∈ℝ{H}.

Substituting in (<ref>) and comparing coefficients gives (i).

ii) Put Ψ := ψ∘ r_h^-1, then Ψ^*ω_0 = (r_h^-1)^*ω_0 and H∘Ψ = H. Therefore by Proposition <ref> we have

    ω_0 - (r_h^-1)^*ω_0 = dH ∧ dη

or again

    r_h^*ω_0 - ω_0 = dH∧ d( 
    dh(H)/dH r_h^*  η).


We want to understand the relationship between α_ω_0,β_ω_0 and α_ω_0,β_ω_0. The Jacobian matrix of the transformation r_h is given by:

    J_r_h = [ 1/2g(H)^-1/2g'(H)∂ H/∂ x x  + g(H)^1/2             1/2g(H)^-1/2g'(H)∂ H/∂ y x;             1/3g(H)^-2/3g'(H)∂ H/∂ x y  1/3g(H)^-2/3g'(H)∂ H/∂ y y + g(H)^1/3 ]

and 

    |J_r_h| =  J_r_h = g(H)^-1/6( g'(H) H + g(H)).

This means that:

    r_h^* ω_0    =α_ω_0(h(H)) |J_r_h|dx ∧ dy + β_ω_0(h(H))| J_r_h|g(H)^1/3ydx ∧ dy + dh(H) ∧ d(r_h^* η).

The last term can be written as dH ∧ d η' with η' = d h(H)/dH r_h^* η. By uniqueness of the characteristic series we get the relations.

iii) Follows from (ii) and Lemma <ref>.



Let's discuss the opposite statement:


 
Consider two  positively-oriented symplectic forms ω_0, ω_0. Suppose there exists a real-analytic map h(H) = H· g(H) such that g(0)>0 and some of the three relations (i), (ii), (iii) of Lemma <ref> is satisfied. Then there exists an H-fibration preserving map ψ such that ψ^*ω_0 = ω_0.


 
We assume (ii) is satisfied, the other two cases are equivalent. Consider the map r_h from the proof of Lemma <ref>. It satisfies r_h^* H = h(H) and (trivially) transforms  ω_0 to r_h^*ω_0, therefore by Lemma <ref>

    α_r_h^*ω_0(H) =  g^-1/6(H)( g'(H) H + g(H))α_ω_0(h(H)) = α_ω_0(H),
    β_r_h^*ω_0(H)=  g^1/6(H)( g'(H) H + g(H)) β_ω_0(h(H)) = β_ω_0(H).

This shows that ω_0 and r_h^*ω_0 have the same charateristic series, therefore

    ω_0 - r_h^*ω_0 = d H∧ dη,

for some real-analytic germ η. As we know from the case of H-preserving maps, this equation implies the existence of a H-preserving diffeomorphism ϕ such that ϕ^*r_h^*ω_0 = ω_0. In conclusion, ψ = r_h ∘ϕ is the map we are looking for.


Finally we show how one symplectic invariant survives in the case of H-fibration-preserving maps. Consider a positively-oriented symplectic form ω_0. Suppose we can solve the equation

    α_ω_0(H)=  K g(H)^-1/6( g'(H) H + g(H)),      K ∈ℝ,

for g(H). Then the rescaling map r_h, with h(H) = H· g(H), transforms H to h(H) and 
the symplectic form ω_0 = (r_h^-1)^*ω_0 to ω_0, where  α_ω_0(H) = K. After this, the other characteristic series β_ω_0(H) survives as an invariant in the usual sense (of H-preserving local diffeomorphisms).


The invariant a_ω_0(H) can be reduced to a constant.


  
By assumption we have a_ω_0(0) > 0. It follows from Corollary <ref> that A_ω_0(0) > 0 as well. Setting g(H) = A_ω_0(H)^6/5 and ω_0=(r_h^-1)^*ω_0, we obtain from Lemma <ref> that A_ω_0(H) = 1. With this choice of h(H) = H· g(H), we have A_ω_0(H) = g(H)^5/6, therefore by Corollary <ref>

    a_ω_0(H)    = A_ω_0'(H)H + 56A_ω_0(H) 
       = 56g(H)^-1/6(g'(H)H +  g(H)),

so that a_ω_0(H) is constant  (and α_ω_0(H) as well).



 
A real-analytic singular Lagrangian fibration with one degree of freedom is symplectomorphic, in a neighborhood of an A_2 singularity, to (one of) the following model:

    H= y^3-x^2,   ω_0= dx∧ dy + f(y^3-x^2) · y  dx∧ dy.

Or, equivalently, in a neighborhood of an A_2 singularity (with one degree of freedom) we can always find local coordinates x and y such that the fibration is defined by the function H=y^3-x^2 and the symplectic structure takes the form ω_0= dx∧ dy + f(y^3-x^2) · y  dx∧ dy.  

In this representation, the real-analytic function f(H) is uniquely defined. □


This proposition says that as a complete symplectic invariant of an A_2 singular fibration with one degree of freedom we may consider one (real-analytic) function in one variable.   Since such a fibration appears as a symplectic reduction of the Lagrangian fibration near a parabolic orbit  (for λ=0), we conclude that parabolic orbits possess non-trivial symplectic invariants and the next section is aimed at describing “all of them”. 








§ PARAMETRIC VERSION
 


Our next step is a parametric version of the above construction.  We now assume that H depends on λ as a parameter:

    H(x,y,λ)= H_λ(x,y)= x^2 + y^3 + λ y

and for each value of λ we consider a symplectic structure ω_λ = f(x,y,λ) dx∧ dy, f>0.


We first give necessary and sufficient conditions for the existence of 
a family of maps ψ_λ from Proposition <ref>.

Following the same idea as before, we choose two 2-dimensional sections 𝒩_1 and 𝒩_2 analogous to the above sections N_1 and N_2 (but now for all values of λ) and define the passage time 

    Π (H,λ) = ∫^𝒩_2(H,λ)_𝒩_1(H,λ)ω_λ/dH_λ
 
for each trajectory
with parameters H and λ, (H,λ)∉Σ_hyp, see  Fig. 2.
Also we see that  for each λ<0 we have a family of closed trajectories also parametrized by H and λ. Let us denote by Π_∘ (H,λ) the period of these trajectories[Alternatively we may compute the area  𝖺𝗋𝖾𝖺_∘(H,λ)=2π I_∘ (H,λ) enclosed by such a trajectory.  This function can be understood as the action variable corresponding to this family of closed cycles.  Notice that Π_∘ and I_∘ are related by differentiation:  Π_∘(H,λ) =  2π∂/∂ H I_∘(H,λ), comp. (<ref>).]. We can compute these functions for both forms ω_λ and ω_λ.   For ω_λ, we denote them by Π(H,λ) and Π_∘(H,λ).








 
A family of local diffeomorphisms ψ_λ from Proposition <ref> exists if and only if 


  (i)
Π(H,λ)- Π(H,λ)  extends to a real-analytic function in a neighborhood of the point  H=0, λ=0, 


  (ii) Π_∘ (H,λ)= Π_∘ (H,λ).




 We need to justify the “if” part only.
First of all we notice that, for each λ  (if we consider each slice {λ=} separately),  a map ψ_λ exists. 
Indeed, for λ>0, there are no obstructions for the existence of ψ_λ at all,
since our fibration is regular.   For λ =0, the existence of ψ_λ was proved in Corollary <ref>.
As for λ <0,  this property follows from non-degeneracy of singular points (see <cit.>).

We only need to “combine” all these maps into one single Ψ(x,y,λ) = ψ_λ (x,y) in such a way that Ψ is real-analytic with respect to all variables (including λ).

To that end, we notice first of all that the maps ψ_λ can be chosen in such a way that each section N_1, λ={ (x,y,λ)∈𝒩_1  }   (i.e. the intersection of 𝒩_1 with the corresponding λ-slice) is mapped to itself, i.e., ψ_λ|_N_1, λ = id.
This choice (of the initial data) makes our construction unique. In other words, we may assume without loss of generality that Ψ leaves 𝒩_1 fixed.  
  
Let σ^t and σ^t denote the Hamiltonian flows of H_λ w.r.t.  ω_λ and ω_λ respectively. Since H is preserved and 
ψ_λ^*(ω_λ)=ω_λ,
we conclude that ψ_λ sends the Hamiltonian flow of H w.r.t ω_λ to that w.r.t. ω_λ, i.e., the following relation holds

    ψ_λ∘σ^t = σ^t ∘ψ_λ.

This relation implies a simple “explicit” formula for ψ_λ (for those points Q which can be obtained from 𝒩_1 by shifting along the flow σ^t).  Namely,  let  Q = σ^t(Q)(Q_0) with Q_0 ∈𝒩_1. Then applying the above relation to the point Q  with  t=-t(Q) we get

    ψ_λ∘σ^-t(Q) (Q) = σ^-t(Q)∘ψ_λ (Q)

or, equivalently,

    ψ_λ (Q) = σ^t(Q)∘ψ_λ∘σ^-t(Q) (Q)

and, using that ψ_λ∘σ^-t(Q) (Q) = ψ_λ  (Q_0) = Q_0 =  σ^-t(Q) (Q), we finally get:

    ψ_λ (Q) = 
    σ^t(Q)∘σ^-t(Q) (Q),

where the time t(Q) is chosen in such a way that   σ^-t(Q) (Q)∈𝒩_1.  Notice that the family ψ_λ so defined automatically satisfies the required conditions (ii) from Proposition <ref>  and is locally analytic w.r.t. all the variables  (including the parameter λ)  everywhere where it makes sense.  The problem, however, is that  (<ref>) works neither at the singular points  nor at the points lying on “small” closed trajectories that appear for λ <0 (the reason is obvious: the Hamiltonian flow σ^t starting from 𝒩_1 does not reach them). 

Below, we will use a slightly different version of formula (<ref>).  Notice that  Q can also be obtained from Q_0∈𝒩_1 by shifting  along the other Hamiltonian flow σ^t, that is, Q= σ^t̃(Q)(Q_0) for some  t̃(Q)∈ℝ.  Hence,

    ψ_λ (Q) = 
    σ^t(Q)∘σ^-t(Q) (Q) =  σ^t(Q)-t̃(Q)∘σ^t̃(Q)∘σ^-t(Q) (Q) =
     σ^t(Q)-t̃(Q)∘σ^t̃(Q) (Q_0)=  σ^t(Q)-t̃(Q) (Q),

or, finally,

    ψ_λ (Q) = σ^r(Q) (Q),


This formula has a very natural meaning.  If Q can be obtained by shifting a certain point Q_0∈𝒩_1  along the flows σ^t and σ^t, then ψ_λ simply shifts Q along σ^t by time  r(Q) = t(Q) - t̃(Q), where t(Q) (resp.  t̃(Q)) is the time necessary for the flow  σ^t (resp. σ̃^t) to reach Q starting from the section 𝒩_1. 


Our goal is to show that this formula extends to a neighborhood of the parabolic point up to a well defined real-analytic map (in the sense of all the variables x,y and λ).




To that end,  we will use a “complexification trick”.   Since all the objects under consideration are real-analytic we can naturally complexify them, that is, we may think of x,y,λ as complex variables, H and F as complex functions, ω and ω as complex symplectic forms, etc.  We will also assume that the section 𝒩_1 is given by an analytic equation like  f(x,y)=0, so that the same equation defines a (local) complex hypersurface that is transversal to all complexified leaves 
ℒ_ε_1,ε_2={H= ε_1, F=ε_2},  (ε_1, ε_2) ⊂ℂ^2, for small enough |ε_1|+|ε_2|.
We are now looking for a local holomorphic map Ψ(x,y,λ)=ψ_λ(x,y) which preserves H and F and transforms 
ω_λ to ω_λ.

We want this map to be the “complexification” of the family ψ_λ defined above  (in particular,  the complex section 𝒩_1 does not move under the action of ψ_λ).  We keep the same notations for all the objects, but  now we think of them from the
complex viewpoint.  In particular,  the parameter t  for the flows σ^t and σ^t is complex and plays the role of “complex time”.   Similarly,  t(Q), t̃(Q) and r(Q) are complex functions which, by construction, are locally holomorphic.   


One of the advantages of the complexified picture is that all the leaves 
ℒ_ε_1,ε_2 
(both regular and singular) are now connected, each of them intersects the section 𝒩_1 at exactly one point and, moreover,  every regular point Q (even if it belongs to a singular leaf)  can be joint with 𝒩_1 by a continuous path lying on the leaf.  Notice that the regular part of each leaf ℒ_ε_1,ε_2
can be understood as a complex trajectory of the complex flow σ^t or σ^t.  

The problem coming with “complexification” is that t(Q) and t̃(Q) are not uniquely defined anymore.  Indeed, the complex leaf 
ℒ_ε_1,ε_2={H= ε_1, F=ε_2}
is now a two-dimensional surface with a non-trivial topology.  In particular, the first homology group of 
ℒ_ε_1,ε_2
is non-trivial and this leads to the fact that  Q can be reached from 𝒩_1 in many different ways,  e.g.,   σ^t_1 (Q_0) = σ^t_2(Q_0) = Q.  So we need to make sure that the choice of t_i does not affect the final result of (the complex version of) (<ref>) and (<ref>).  

Let us discuss this issue in more detail.  Consider one particular leaf 
ℒ_ε_1,ε_2 (not necessarily regular).
It intersects the section 𝒩_1 at exactly one point Q_0.    For 
Q∈ℒ_ε_1,ε_2,
consider a path γ(s) connecting this point with Q_0 so that γ (0)= Q_0 and γ(1)= Q.    Each point of this path can be written as γ(s) = σ^t (s) (Q_0) = σ^t̃ (s) (Q_0)  with t(s)∈ℂ, t(s) continuous and t(0)=0.  In this way, we set t(Q)=t(1) 
and
t̃(Q) = t̃(1). It is easy to see that deforming γ(s) continuously does not change 
t(Q) and t̃(Q).  Thus, this construction shows that t(Q) and t̃(Q)  (and consequently r(Q)=t(Q)-t̃(Q)) are uniquely defined if we fix the homotopy type of a curve connecting Q_0 and Q.  If we choose two homotopically different curves γ_1 and γ_2, then,  in general,  t_1(Q) t_2(Q) and t̃_1(Q) t̃_2(Q).  

The condition we need is  r_1(Q) = t_1(Q)-t̃_1(Q) = t_2(Q)-t̃_2(Q) = r_2(Q) or, equivalently,  t_1(Q)-t_2(Q) = t̃_1(Q)- t̃_2(Q).  The latter has a very simple geometric meaning. Namely,  Q= σ^t_1(Q) (Q_0) = σ^t_2(Q) (Q_0) means that 
σ^t_1(Q)-t_2(Q) (Q)=Q.  In other words, t_1(Q)-t_2(Q)  is the period of 
ℒ_ε_1,ε_2
as a “complex trajectory” of the flow σ^t, which corresponds to the (homotopy class of the) 
loop formed by the curves γ_1 and -γ_2.  
In other words, the condition that we need can be formulated as follows:  for each 
loop γ on ℒ_ε_1,ε_2,
the corresponding periods of the Hamiltonian flows generated by H_λ  w.r.t. the symplectic forms  ω_λ and ω_λ coincide.  These periods can be computed explicitly as  (compare with (<ref>)):

    Π_γ(H,λ) = ∮_γω_λ/ dH_λ    Π_γ(H,λ) = ∮_γω_λ/ dH_λ,

so that the required condition takes the following form:

    Π_γ(H,λ)  =  Π_γ(H,λ)

for any closed loop γ on ℒ_ε_1,ε_2 = ℒ_H,λ  (equivalently, for any cycle of the first homology group).

Let us assume that this condition holds true  (we will below explain why, under our assumptions, this is indeed the case)  and make the next step of our construction.  As just shown,  (<ref>) guarantees that  the function r(Q) is well defined for any point Q that can be reached by the flows 
σ^t and σ^t starting from the section 𝒩_1.    Since (after complexification!) every regular point satisfies this property, r(Q) is defined everywhere except for singular points and is locally holomorphic by construction.  But the set of singular points, 

    {∂ H/∂ x=0, ∂ H/∂ y=0}
    ={x=0, 3y^2+λ=0},
 
is an algebraic variety of (complex) codimension 2, and therefore by the second Riemann extension theorem (<cit.> or <cit.>), 
r(Q) can be extended up to a holomorphic function defined everywhere in the considered domain.   In particular, this function is bounded and therefore,  by taking a smaller neighborhood of the parabolic point,  we may assume that the flow σ^t is well defined for all  t satisfying |t|≤max |r(Q)|. 



After this, our formula
(<ref>) can 
be applied to every point from this neighborhood giving a well defined holomorphic map Ψ with required properties.  It remains to return to the real world  (i.e., restrict ψ_λ to the real part of our complex neighborhood) and we are done.


To complete the proof we need to explain why condition (<ref>) is fulfilled in our case.  First we notice that the first homology group of complex leaves ℒ_ε_1,ε_2 is generated by 2 cycles  (topologically, ℒ_ε_1,ε_2 is a torus with one hole  if (ε_1,ε_2)∉Σ^ℂ={ε_1^2=-4/27ε_2^3}, a 2-disk with one hole if (ε_1,ε_2)=(0,0), and a pinched torus with one hole otherwise, where one of the basic cycles is pinched to a point).  Consider the (real) 
“swallow-tail domain” {(ε_1,ε_2)∈^2|ε_1^2<-4/27ε_2^3}⊂{λ <0}.  Then one of these two cycles can be chosen real. Such a cycle is shown in Fig. 2 as a small 
loop, whose periods w.r.t.  ω_λ and ω_λ  were denoted by  Π_∘ (H,λ) and Π_∘ (H,λ).  By our assumption Π_∘ (H,λ)=Π_∘ (H,λ), i.e., one of the required conditions  coincides with the second condition (ii) of Proposition  <ref>.   

Now consider  condition (i)  for {(ε_1,ε_2)∈^2|ε_1^2<-4/27ε_2^3}⊂{λ <0}.   When approaching a hyperbolic singular leaf, the  functions Π(H,λ) and Π(H,λ) both have logarithmic singularity.  This is a well known property of non-degenerate hyperbolic points (<cit.>),  in other words,  they have the following asymptotics[In the domain {(ε_1,ε_2)∈^2|ε_1^2>-4/27ε_2^3}, similar asymptotics for Π(H,λ) and Π(H,λ) hold, where the coefficients α,β,α,β are replaced by 2α,δ,2α,δ for some real-analytic functions δ,δ in a neighbourhood of Σ_hyp.]:

    Π(H,λ)    =  α(H,λ) ln| 
    3√(3) H - 2(-λ)^3/2| + β(H,λ), 
    Π(H,λ)    =  α(H,λ) ln| 
    3√(3) H - 2(-λ)^3/2| + β(H,λ)

for some real-analytic functions α, β, α, β in a neighborhood of Σ_hyp=Σ∩{H>0}. Condition (i) of  Proposition   <ref>, therefore, implies that α(H,λ) = α(H,λ).  For  hyperbolic points, this coefficient in front of logarithm is known to be proportional to the period of the second  (invisible in the real setting) cycle on the complex leaf ℒ_H,λ  (see Proposition <ref> and discussion in Appendix). 

Thus, for real λ <0 and real H ∈(-2(-λ)^3/2/(3√(3)),2(-λ)^3/2/(3√(3))) the required conditions (<ref>) are fulfilled.  Since the periods Π_γ and Π_γ are locally holomorphic (we cannot consider them as single-valued functions because of the monodromy phenomenon) and coincide on an open real domain, we conclude that (<ref>) is fulfilled identically, which completes the proof of Proposition <ref>.



We now return to our discussion on symplectic invariants of parabolic trajectories that we started in Section <ref>.   According to Proposition 
<ref>,  this problem can be reduced to the situation explained in Remark <ref>. 

Namely, we consider two functions H=x^2+y^3+λ y and F=y that commute simultaneously with respect to two symplectic forms Ω and Ω defined by (<ref>) and (<ref>)  with ω_λ = f(x,y,λ) dx∧ dy and  ω̃_λ = f̃(x,y,λ) dx∧ dy and f, f̃ >0. Combining Proposition <ref> and Proposition <ref>, we obtain the following 


The following two statements are equivalent.


  * In a tubular neighborhood of the parabolic orbit γ_0(t)=(0,0,0, φ=t) there is a (real-analytic) diffeomorphism Φ  such that



  (i) Φ preserves H and F;

  (ii) Φ^* (Ω) = Ω.



  * The functions Π,Π_∘,Π,Π_∘ (real-analytic in the complement of the bifurcation diagram) satisfy the relations


  * Π(H,λ)- Π(H,λ)  is real-analytic    (in a neighborhood of the point  H=0, λ=0), 


  * Π_∘ (H,λ)= Π_∘ (H,λ).
□






In fact, the functions Π(H,λ) and Π_∘ (H,λ) are not independent.  Indeed, as  
H→  2(-λ)^3/2/(3√(3)) 
(i.e., when the real disconnected regular fiber approaches the hyperbolic singular one) these two functions  have a logarithmic singularity with the same logarithmic coefficient, that is, we have the following asymptotics:

    Π(H,λ)    =  α(H,λ) ln| 
    3√(3) H - 2(-λ)^3/2| + β(H,λ), 
    Π_∘(H,λ)    =  α(H,λ) ln| 
    3√(3) H - 2(-λ)^3/2| + β_∘(H,λ).
  
In other words, the functions β(H,λ) and β_∘(H,λ) are different and not related to each other in any sense,  but 
the coefficients α(H,λ) are the same for the both functions.   According to Proposition <ref>, however,   the regular part  β(H,λ) of Π(H,λ) does not play any role, so that the only important information for us is the coefficient α(H,λ) which, as we have just explained,  can be “obtained” from Π_∘(H,λ).  Hence we conclude that   
Π_∘(H,λ) contains all the information we need for symplectic characterisation of a parabolic trajectory.  

We also note  that the period Π_∘(H,λ) of closed trajectories can naturally be interpreted in terms of the action variables of our integrable system. Indeed, the family of small closed trajectories shown on Fig. 2 corresponds to a family of 
“narrow”
two-dimensional Liouville tori  (recall that a four-dimensional neighborhood U(γ_0) of the parabolic orbit γ_0 is the product (Fig. 2)× S^1). For this family, we can naturally define two action variables I_1 and I_2. The first of them corresponds to 
the free Hamiltonian S^1-action on U(γ_0) generated by F=λ, that is, I_1 = λ. The other I_2(H,λ)  corresponds to the family of vanishing cycles shown in Fig. 2  as small closed trajectories. 
We re-denote this function as I_2(H,λ)=I_∘ (H,λ).
Without loss of generality  we will assume  that

    I_∘>0      I_∘→0      (H,λ)→ (H(γ_0),F(γ_0)),

i.e., as we approach the singular fiber.  Notice that,  in a coordinate system (x,y,λ,φ), I_∘ (H,λ) can be defined  by an explicit formula.  
Fixing H and λ, we define a unique closed cycle (see Fig. 2). This cycle bounds a certain domain V_H,λ⊂^2 (x,y) on the corresponding layer {λ = const}.  Then

    I_∘ (H,λ) = 1/2π𝖺𝗋𝖾𝖺_∘(V_H,λ) =   1/2π∫_V_H,λω_λ.



It is well-known that I_∘ (H,λ) and Π_∘(H,λ)>0 are related in the following very simple way:

    Π_∘(H,λ) = 2π∂/∂ HI_∘ (H,λ),

which shows that Π_∘(H,λ) can be reconstructed from I_∘ (H,λ), so that we finally come to the following equivalent  version of 
Proposition <ref>.




  
In the same assumptions as in Proposition <ref>, the following two statements are equivalent.


  (i) In a tubular neighborhood of the parabolic orbit γ_0 there is a (real-analytic) diffeomorphism Φ  such that


  * Φ preserves H and F;

  * Φ^* (Ω) = Ω.


  (ii)  The actions (real-analytic on the 
“swallow-tail domain” corresponding to a family of “narrow” Liouville tori)
corresponding to the family of vanishing cycles 
(cf. (<ref>))
coincide, I_∘(H,F)= I_∘ (H,F).   
□





We now want to give one more version of the criterion for the existence of Φ by omitting the condition F=λ which, in particular, means that F is a 2π-periodic integral (equivalently, the action variable I_1) simultaneously for the both integrable systems.   


Consider  H and F commuting with respect to Ω and Ω in a tubular neighborhood of a parabolic orbit γ_0.  Notice that now we are not allowed to assume that these two integrable systems share the same canonical coordinate system (x,y,λ,φ)  as we did in Propositions <ref> and <ref>. 


Let dF|_γ_0 0. We will say that Ω and Ω induce


  * the same orientation of γ_0 if the Hamiltonian flows of F w.r.t. Ω and Ω induce the same orientation of γ_0;

  * the same coorientation of γ_0 if  (the restrictions of) 
Ω and Ω induce the same orientation of a (local) 2-dimensional surface in {F=F(γ_0)} transversal to γ_0 (i.e., on a 2-dim Poincaré section).



Without loss of generality, we may (and will) assume that Ω and Ω induce the same orientation and the same coorientation of γ_0. Indeed, we can easily achieve this condition  by using additional maps  (x,y,λ,φ) ↦  (x,y,λ,-φ) and  (x,y,λ,φ) ↦  (-x,y,λ,φ) (written in a canonical coordinate system from Proposition <ref>) that change respectively the orientation and coorientation without changing the functions F and H.


As above we can define two natural action variables for each of these two integrable systems  I(H,F), I_∘(H,F) and I(H,F), I_∘(H,F). Here I(H,F) and I(H,F) are smooth on a certain neighborhood U(γ_0) and  are generators of  the Hamiltonian S^1-actions w.r.t. Ω and Ω respectively.  

Alternatively, we may define I(H,F)  by 

    I(H,F) = 1/2π∮_γϰ,    dϰ = Ω
 
and γ=γ_H,F is a closed cycle on the fiber ℒ_H,F that is homotopic to γ_0  (recall that locally our fibration  can be understood as the direct product  of S^1 and a three-dimensional foliated domain 𝒱 shown in Figure 2,   then  γ_H,F can be taken of the form S^1 ×{P} where P∈𝒱 is a point lying on the corresponding fiber).


The other action variable I_∘(H,F) is only defined on the family of “narrow” Liouville tori  corresponding to small oriented loops  μ_∘ = μ_∘ (H, F) shown in Fig. 2:

    I_∘ (H,F)= 1/2π∮_μ_∘ϰ ,   
    dϰ = Ω.

In other words, I_∘(H,F) is a function defined on the “swallow-tail” domain on ^2(H,F) bounded by the bifurcation diagram Σ (this definition coincides with (<ref>) up to, perhaps, changing the sign).


The actions I(H,F) and I_∘(H,F) for the second system are defined in a similar way by integrating ϰ,  dϰ = Ω,   over the same cycles γ and μ_∘ with the same orientations.


 
Suppose that the singular fibration defined by the functions H and F is Lagrangian w.r.t. both the symplectic forms Ω and Ω. Suppose that Ω and Ω induce the same orientation and the same coorientation of a parabolic orbit γ_0.
Then the following two statements are equivalent.




  (i) In a tubular neighborhood of the parabolic orbit γ_0 there is a (real-analytic) diffeomorphism Φ  such that



  * Φ preserves H and F;

  * Φ^* (Ω) = Ω.



  (ii)  These two integrable systems have common action variables described above, i.e.,    

    I (H,F)= I (H,F) +        
    I_∘(H,F)= I_∘ (H,F).
  






Suppose (ii) holds true. First of all we replace the functions F and H by new functions F̂  and Ĥ  satisfying the following conditions  (cf. Proposition  <ref> and Remark <ref>):



  * F̂ = ± I(H, F) + const  where ± and const are chosen in such a way that  F̂=0 on the parabolic trajectory γ_0 and F̂<0 on the swallow-tail domain of the bifurcation diagram;


  * Ĥ is chosen in such a way that the bifurcation diagram of  ℱ̂=(F̂, Ĥ) takes the standard form (<ref>).    
   


After this we apply  Proposition <ref> which says that formulas from Proposition <ref> holds true exactly for the functions F̂ and Ĥ.   In other words, we can introduce two different “good”  coordinate systems (x,y,λ, φ) and (x̃, ỹ, λ̃, φ̃)  as in  Proposition <ref> for (Ĥ, F̂, Ω) and  (Ĥ, F̂, Ω) respectively  (notice that  λ = λ̃ automatically  as both λ and λ̃ coincide with F̂).   

The next step is to consider the map Ψ: (x,y,λ, φ) ↦ (x̃, ỹ, λ̃, φ̃) and after this continue working with the forms Ω and Ψ^*(Ω).  Now (x,y,λ, φ) is a common “good” coordinate system for the both systems and the conditions of Theorem <ref> are still fulfilled for 
Ω and Ψ^*(Ω).   
After this,  it remains to apply Proposition <ref>  for the integrable systems (Ĥ, F̂, Ω) and (Ĥ, F̂, Ψ^*(Ω)).

The fact that (i) implies (ii) follows from the assumption that the symplectic forms Ω and Ω induce the same  orientation and co-orientation on γ_0. Indeed this  implies that Φ preserves the homology class of γ and μ_∘ on each “narrow” torus. Therefore  if we set ϰ = Φ^*ϰ in the definition of the actions I(H,F) and I_∘(H,F), then I (H,F)= I (H,F) and I_∘(H,F)= I_∘ (H,F).


In fact, we do not even need to mention H and F in the statement of Theorem <ref> at all.  We may simply say:   


 
Consider a singular fibration with a parabolic orbit γ_0 which is 
Lagrangian with respect to two symplectic structures Ω and Ω.
Suppose that Ω and Ω induce the same orientation and the same coorientation of γ_0.
The necessary and sufficient condition
for the existence of a (real-analytic) diffeomorphism Φ in a 
tubular
neighborhood of γ_0 sending each fiber to itself
and such that Φ^* (Ω) = Ω is that these two systems have common action variables in the sense that
for every closed cycle τ on any “narrow” torus   
we have  

    ∮_τϰ =  ∮_τϰ,     
    dϰ = Ω,   dϰ= Ω,

where ϰ and ϰ are chosen in such a way that ∮_γ_0ϰ = ∮_γ_0ϰ=0.□


Notice that due to analyticity it is sufficient to compare  the actions only on the family of “narrow” tori,  although I and Ĩ are defined on the whole neighborhood U(γ_0).



Finally, we want to relax the condition that each fiber goes to itself  (indeed, this assumption makes no sense at all if we want to compare parabolic orbits for two different integrable systems).   





Assume that we are given two integrable systems with parabolic orbits γ_0 and γ_0, respectively. For the both systems we consider the bifurcation diagrams (or bifurcation complexes), Σ and Σ respectively, and the 
“swallow-tail domains” corresponding to the families of “narrow” Liouville tori.  On each of these domains
we have two actions I and I_∘  (as functions of H and F) and correspondingly I and I_∘  (as functions of H and F) defined as above.  Without loss of generality we will assume that these action variables are “normalised” in such a way that 


  * all of them vanish at  the corresponding cusp point, 

  * I_∘ and I_∘ are positive on the corresponding “swallow-tail” domains,

  * I and Ĩ are negative on the corresponding “swallow-tail” domains.

Combining Theorem <ref> with Proposition <ref>  we obtain

 
The necessary and sufficient condition for the existence of a real-analytic fiberwise symplectomorphism Φ: U(γ_0) →U (γ_0) between some tubular neighborhoods U(γ_0),U (γ_0) of the parabolic orbits γ_0,γ_0 is that these two systems have common action variables in the sense that 
there is a real-analytic diffeomorphism

    ϕ: (H, F) ↦ (H, F)

between some neighborhoods of the cusp points (H(γ_0),F(γ_0)) and ( H(γ_0), F(γ_0)) in ^2
that 


  * respects the bifurcation diagrams together with their partitions into hyperbolic and elliptic branch[Equivalently, we may say that ϕ  defines a (local) homeomorphism between the corresponding bifurcation complexes.]:

    ϕ(Σ)=Σ,     ϕ(Σ_ell)=Σ_ell    ϕ(Σ_hyp)=Σ_hyp,


  * and preserves the action variables described above: I= I∘ϕ and I_∘= I_∘∘ϕ, i.e., for the action variables defined on the “swallow-tail domains” we have 

    I(H,F)=I (H(H,F), F(H,F))      I_∘(H,F)=I_∘ (H(H,F), F(H,F)).
    □







The latter conclusion basically means that the only symplectic invariants of hyperbolic orbits are action variables.  This conclusion does not provide any tools to decide whether a suitable map (<ref>) (making the actions equal) exists or not, but some necessary conditions can be easily found.   Some of them have
been already described in Section <ref>,  e.g., the function f(·) from Proposition <ref>.  This function is a symplectic invariant of a parabolic singularity which “corresponds” to the level λ =0, where
λ, as above, denotes the first action variable I(H,F).  We now want to describe another non-trivial symplectic invariant which will be a function h(λ), λ <0.


Since λ=λ(H,F) is a real-analytic function, we can consider it as a parameter on the hyperbolic branch Σ_hyp of the bifurcation diagram Σ.   Consider  I_∘ (H,λ) as a function of H (with λ as a parameter). This function is defined on the interval   

    (-2(-λ)^3/2/(3√(3)),2(-λ)^3/2/(3√(3))),

is strictly increasing from 0 to its maximum attained on the hyberbolic branch. We denote it by  h(λ)=max_H I_∘ (H,λ).  Obviously, h(λ)  does not depend on the choice of commuting functions H and F defining the Lagrangian fibration, so that  h(λ) can be considered as a symplectic invariant of a parabolic singularity. 

The problem of an explicit description of a complete set of symplectic invariants
is equivalent,  as shown above, to the analysis of the asymptotics of the function I_∘ (H,λ). More precisely,  we should describe invariants of such functions under (real-analytic) transformations of the form (H,λ)  ↦( H (H,λ), λ= λ).  







§ SEMI-LOCAL  INVARIANTS OF CUSP SINGULARITIES
 


Finally, we want to describe semi-local invariants of cusp singularities.  In other words, we now consider a saturated neighborhood of a compact singular fiber ℒ_0 containing a parabolic orbit, i.e., cuspidal torus.  We assume that this fiber  contains no other critical points, so that the topology of the fibration in a neighborhood of ℒ_0 is  standard and illustrated in Figure 3.   This Figure also shows the bifurcation complex, i.e., the base of this fibration, which consists of two  2-dimensional strata (attached to each other along Σ_hyp,  one of the branches of the bifurcation diagram Σ that corresponds to the family of hyperbolic orbits).  Each stratum represents a family of Liouville tori and therefore we can naturally assign a pair of action variables to each of them.  Our goal is to show that fibrations with the same actions are symplectomorphic.  

 In a neighborhood  U(ℒ_0) of the singular fiber ℒ_0,  on all neighboring Liouville tori we can choose a natural basis of  cycles in the first homology group of H_1(T^2_F,H, ℤ) 
where T^2_H,F is the Liouville torus defined by fixing the values of the integrals F and H respectively.   These cycles are shown in Figure 3.  One of them corresponds to the S^1-action defined on U(ℒ_0)  (in Figure 3, this cycle γ is denoted by S^1).  The other cycle can be obtained by considering a global 3-dimensional cross-section to this S^1-action.  Since this S^1-action  (and the corresponding S^1-fibration) is topologically trivial, such a cross section exists.  It is illustrated on the left in Figure 3 and denoted by V  so that we may think of  U(ℒ_0) as the direct product V × S^1 = U(ℒ_0).    Each Liouville torus  T^2_F,H intersects V along a closed curve  (these curves are shown in Figure 3) and this curve is taken as the second basis cycle μ in H_1(T^2_F,H, ℤ).  More precisely, we need to take into account that for a point (F,H) from the swallow-tail zone,  we will have two disjoint Liouville tori.   The corresponding cycles will be denoted by μ and μ_∘, where μ_∘ is used for the vanishing cycle on the family of “narrow” tori, the other, i.e., μ,  corresponds to a “wide” torus. 

Notice that the first cycle γ is uniquely defined by the S^1-action. The cycle μ_∘ is also well defined by the topology of the fibration  (as a vanishing cycle). The other cycle μ is not.  It is easy to see that μ is defined up to the transformation of the form μ↦μ + kγ,   k∈ℤ.   This is caused by ambiguity in the choice of the cross-section V which can be chosen in many homotopically different ways (this phenomenon is discussed  and explained in details in <cit.>).  

Summarizing,  on each stratum of the bifurcation complex, we have a pair of action variables  I_γ, I_μ  and I_γ, I_μ_∘ (the latter for the swallow-tail stratum).   Each of these functions can be treated as a real-analytic function of H and F.  In fact,  we have already considered the actions I_γ and I_μ_∘ in the previous Section <ref>, where they were denoted by I(H,F) and I_∘ (H,F).   We will keep this notation here, i.e., we set  I_γ = I, I_μ_∘ = I_∘. The remaining action will be denoted by I_μ so that we have 3 action variables I, I_∘ and I_μ.  The first two of them are well-defined,  but I_μ is defined modulo transformation I_μ↦ I_μ + k I. 

Also notice that I(H,F) is real-analytic everywhere  (strictly speaking we need to distinguish this action for the families of “narrow” and “wide” tori, but due to real-analyticity  I(H,F), as function of H and F,  is the same for the both families).  The function  I_μ (H, F)  is defined and is real-analytic everywhere except for the hyperbolic branch Σ_hyp of the bifurcation diagram.  When approaching  Σ_hyp the function tends to certain finite limits, but these limits from above and from below are different.   The function I_∘ is defined on the swallow-tail domain and is continuous on  its closure.

10pt







Our final result basically states that the systems with equal actions are symplectomorphic.  We will give two versions of this result.
Consider two integrable Hamiltonian systems (H,F,Ω, U(ℒ_0)) and (H,F,Ω, U(ℒ_0)) defined on some neighborhoods[We do not specify the sizes of these neighborhoods, but assume that they are sufficiently small. In other words, we are talking about germs of fibrations and germs of maps.] of cuspidal tori ℒ_0 and ℒ_0.




Assume that there is a fiberwise diffeomorphism Ψ : U(ℒ_0) →U(ℒ_0) that preserves the actions in the sense that for every cycle τ⊂ℒ_H,F we have

    ∮_Ψ(τ)ϰ = ∮_τϰ
 
for some 1-forms ϰ and ϰ satisfying dϰ = Ω, dϰ=Ω.
Then there exists a fiberwise symplectomorphism  Φ: U(ℒ_0)→U(ℒ_0). 


The converse statement is obviously true, since a fiberwise symplectomorphism Φ preserves the actions:

    ∮_Φ(τ)ϰ = ∮_τϰ
 
where ϰ and ϰ are related by Φ^*ϰ = ϰ.

 

A stronger version is as follows.  
For each system we compute the actions I(H,F), I_∘(H,F) and I_μ (H,F) and respectively  I(H,F), I_∘(H,F) and I_μ (H,F)  for the second system as explained above.   




 Assume that there is a local real-analytic diffeomorphism  ϕ : (H,F) ↦ ( H,  F),  H= H(H,F) and F= F(H,F)
that 


  * respects the bifurcation diagrams together with their partitions into hyperbolic and elliptic branches:

    ϕ(Σ)=Σ,     ϕ(Σ_ell)=Σ_ell    ϕ(Σ_hyp)=Σ_hyp,


  * and makes the actions equal (for some choice of μ and μ):

    I (H,F)    =  I ( H(H,F),  F(H,F)), 
    
    I_∘ (H,F)    =  I_∘ ( H(H,F),  F(H,F))  
    
    I_μ (H,F)    =  I_μ  ( H(H,F),  F(H,F)).


Then there exists a fiberwise symplectomorphism  Φ: U(ℒ_0)→U(ℒ_0).



Notice that the converse statement is also true: a fiberwise symplectomorphism Φ:U(ℒ_0)→U(ℒ_0) induces a diffeomorphism ϕ between the bases of the fibrations which automatically satisfies the properties above (where the choice of μ is induced by Φ and μ).



We can rewrite this statement in a slightly different and shorter way.     For each of the above integrable systems we may consider the momentum map π :  U(ℒ_0) → B ⊂^2(H, F) and π :  U(ℒ_0) →B⊂^2(H, F),  where B and B are some neighborhoods of the corresponding cusp points of the bifurcation diagrams.  Then we can think of the actions as functions on B (more precisely on the corresponding domains defined by the bifurcation diagrams).   Then Theorem <ref> can be rephrased as follows:  


Assume that there exists a local real-analytic diffeomorphism ϕ : B →B respecting the bifurcation diagrams Σ and Σ and such that 
I = I∘ϕ,  I_∘ = I_∘∘ϕ and I_μ = I_μ∘ϕ. Then there exist a fiberwise symplectomorphism  Φ: U(ℒ_0)→U(ℒ_0).






The proof of this theorem is based on the following lemma. Consider two (non-singular) integrable systems (H, F, Ω) and (H, F, Ω) defined in some neighborhoods  T^2 × B and  T^2×B of regular Liouville tori. Here B and B̃ are 2-dimensional discs viewed as the bases of the corresponding (regular) Lagrangian fibrations  endowed with induced integer affine structures   (action variables).  The functions (H,F) and (H, F) are treated as smooth functions  on B and B respectively. We also consider the Hamiltonian ^2-actions  σ^(t_1, t_2) and σ^(t_1, t_2), (t_1, t_2)∈^2 generated by the commuting functions (H, F) and (H̃, F̃).  Here σ^(t_1, t_2) denotes the composition of the Hamiltonian shifts along vector fields X_H and X_F by time t_1 and time t_2 respectively.  Similarly for σ̃^(t_1, t_2). 



Assume that we have a real-analytic diffeomorphism  

    ϕ : B →B̃,
  
which provides an (integer) affine equivalence between B and B.   Let   H =  H ∘ϕ and  F =  F ∘ϕ   and consider two Liouville tori T_p =T^2 ×{p}   and  T_ϕ(p) = T^2 ×{ϕ (p)} where p∈ B is an arbitrary point (in other words, these tori correspond to each other under the map ϕ: B →B).    Let x∈ T_p and x̃∈T_ϕ(p) be arbitrary two points from these fibers.   

Then σ^(t_1,t_2) (x) = x   (or more generally σ^(t_1, t_2)(x) = σ^(t'_1, t'_2)(x))  if and only if 
σ^(t_1,t_2) (x̃) = x̃  (respectively σ^(t_1, t_2)(x̃) = σ^(t'_1, t'_2)(x̃)). 


  We will give a proof of this statement in the case of n degrees of freedom.   Recall that B and B are endowed with integer affine structures induced by the action variables.  By definition,  ϕ: B →B is an  (integer)  affine equivalence if ϕ sends “actions to actions”. More precisely,
let I_1, …,  I_n  be action variables for B̃,  which means that these functions define the Hamiltonian action of the standard torus ^n/Γ_0 where Γ_0=ℤ^n is the standard  integer lattice in ^n. 
We say that ϕ :B → B is an affine equivalence, if I_1= I_1 ∘ϕ, … , I_n= I_n ∘ϕ are action variables on B.

If in Lemma  <ref> instead of (H, F) and (H, F) we consider (I_1, I_2) and ( I_1,  I_2), then the statement  is obvious:   both relations σ^(t_1, t_2) (x) = x and σ^(t_1, t_2) ( x) =  x  simply mean that (t_1, t_2) belongs to the standard integer lattice, i.e., t_1, t_2 ∈ℤ.


Let us see what happens if take arbitrary functions  (H, F)  or, more generally, (F_1, F_2, … F_n) in the case of n degrees of freedom.  The relation σ^(t_1, …, t_n) x = x  means that  (t_1,…, t_n) belongs to the period lattice Γ⊂^n which is the stationary subgroup of x in the sense of the Hamiltonian ^n-action generated by F_1, F_2, … F_n.   Since this lattice is the same for any point x from a fixed torus T_p, p ∈ B,  we may denote it by Γ(T_p).   This lattice is not standard anymore and it depends on two things,  the torus  T_p  (or just a point p∈ B) and the generators F_1, F_2, … F_n of the Hamiltonian ^n-action.

If we know the expressions of F_1, …, F_n in terms of the actions I_1,…, I_n, then the lattice Γ(T_p) is easy to describe. Namely: 

    Γ(T_p) =  Γ_0 · J^-1(p),

where Γ_0 is the standard integer lattice and J(p) denotes the Jacobi matrix  J(p) = (  J^i_j= ∂ F_i/∂ I_j|_p).  In more details,

    (t_1, … , t_n) ∈Γ(T_p)      (t_1, … , t_n) = (k_1,…, k_n) · J^-1(p)

for (k_1,…, k_n)∈Γ_0,  i.e., for some vector with integer components k_i  ∈ℤ. 

The same, of course, holds for x ∈ T_ϕ(p),  that is 

    (t_1, … , t_n) ∈Γ( T_ϕ(p))      (t_1, … , t_n) = (k_1,…, k_n) · J^-1(ϕ(p))

where J(ϕ(p)) = (  J^i_j= ∂ F_i/∂ I_j|_ϕ(p)).  It remains to notice that under our assumptions these matrices coincide.   The reason is obvious:  since I_k =  I_k ∘ϕ and also F_i =  F_i ∘ϕ,  we see that 
F_i = f_i(I_1,…,I_n) implies that F = f_i ( I_1, … ,  I_n), i.e.,   F_i depends on I_1,…,I_n  exactly in the same way as 
F_i depends on I_1,…,  I_n so that the corresponding partial derivatives (being computed at p and ϕ(p), i.e., at those points for which (I_1,…,I_n) = ( I_1,…,  I_n)) obviously coincide. In other words, we have proved that Γ(T_p) = Γ ( T_ϕ(p)), which is equivalent to our statement. 










This lemma implies the following two extension results.


Under the assumptions and notation from Lemma <ref>, assume that N and N are Lagrangian (real-analytic) sections of the Lagrangian fibrations  π: T^2 × B → B and π: T^2 ×B→B respectively.   Since the sections N and N can be naturally identified with the bases B and B,  the map ϕ : B →B  induces a natural map between N and Ñ which we denote by the same letter ϕ : N →N.  For any point y∈ T^2 × B we can find  (not uniquely!)  (t_1(y), t_2(y))∈^2 such that x=σ^(t_1(y), t_2(y))(y) ∈ N.  Consider the map Φ: T^2 × B →T^2×B defined by

    Φ (y) = σ^(-t_1(y), -t_2(y))(ϕ(x)),     x=σ^(t_1(y), t_2(y))(y) ∈ N.


 

The map Φ (y) is well defined  and is a fiber-wise real-analytic diffeomorphism  satisfying Φ^*(Ω) = Ω.


 The fact that Φ is well defined (i.e., does not depend on the choice of (t_1, t_2)∈^2  with the property σ^(t_1, t_2)(y) ∈ N) follows from Lemma <ref>.  To show that Φ is symplectomorphism, i.e., Φ^*(Ω) = Ω,  we notice that the position of each point y ∈ T^2 × B is defined by the values of H, F   (which can be understood as coordinates on B) and t_1, t_2 (which can be understood as coordinates on the torus T^2 with the “origin” (0,0) located on N).  These four functions define a canonical coordinate system, i.e.,

    Ω = dH∧ dt_1 + dF∧ dt_2.

A similar canonical coordinate system  H̃, F̃, t̃_1, t̃_2 can be defined on T^2 × B by using the action σ and the Lagrangian section N.  It remains to notice that our map Φ in these coordinate systems, by construction, takes the form
H = H, F = F, t_1=t_1, t_2 = t_2.  





Let U⊂ T^2 × B be an open subset such that the intersection of U with each fiber is connected and non-empty.  Let  Φ_loc :  U →U be a real-analytic fiber-wise diffeomorphism with a certain open subset U⊂T^2 ×B such that Φ_loc^*(Ω) = Ω.  Since Φ_loc is fiberwise and U intersects each fiber,   Φ_loc induces a real-analytic map ϕ between the bases B and B.  


Φ_loc can be extended  up to 
a real-analytic fiber-wise diffeomorphism  Φ: T^2 × B →T^2 ×B with the property Φ^*(Ω) = Ω   if and only ϕ: B →B is an integer affine equivalence.  







   First of all we notice that such an extension (if it exists) is always unique.  Indeed,   since Φ is a symplectomorphism,  we have

    Φ∘σ^(t_1, t_2) =  σ^(t_1, t_2)∘Φ,

where σ and σ are Hamiltonian ^2-actions generated by H,F  and  H=H∘Φ^-1, F= F∘Φ^-1 respectively.   Therefore for any y ∈ T^2 × B,   its image Φ (y) is uniquely defined by:

    Φ(y) = σ^(t_1, t_2)∘Φ_loc∘σ^(-t_1, -t_2) (y),

where (t_1, t_2) are chosen in such a way that σ^(-t_1, -t_2) (y)∈ U  (such (t_1, t_2)∈^2 exists as each orbit of the action σ has a non-trivial intersection with U).   Moreover,  this formula can be understood as an explicit formula for the required extension.  
In a neighborhood of every point y,  the expression σ^(t_1, t_2)∘Φ_loc∘σ^(-t_1, -t_2)  (with fixed (t_1, t_2)) is a composition of three real-analytic fiberwise symplectomorphisms.  So the only condition we need to check is that formula (<ref>) is well defined, i.e.,  does not depend on the choice of (t_1, t_2)∈^2. 

Assume that

    y = σ^(t_1, t_2) (x) = σ^(t'_1, t'_2) (x')    x,x'∈ U.


We need to check that

    σ^(t_1,t_2)∘Φ_loc∘σ^(-t_1,-t_2) (y) = 
    σ^(t'_1,t'_2)∘Φ_loc∘σ^(-t'_1,-t'_2) (y)

or, equivalently,

    σ^(t_1,t_2)∘Φ_loc (x) = 
    σ^(t'_1,t'_2)∘Φ_loc (x').

By our assumption,  the intersection of U with each torus (interpreted now as an orbit of σ) is connected,   therefore there exists a continuous curve (ε_1(s), ε_2(s)),  s∈ [0,1] and  ε_1(0)= ε_2(0)=0 such that

    σ^(ε_1(s), ε_2(s)) (x) ∈  U    s∈[0,1]     σ^(ε_1(1), ε_2(1)) (x)= x'.

Since Φ_loc is a fiberwise symplectomorphism, we have 

    Φ_loc∘σ^(ε_1(s), ε_2(s)) (x) =  σ^(ε_1(s), ε_2(s))∘Φ_loc (x)

for any s and, in particular,

    Φ_loc (x') =  σ^(ε_1(1), ε_2(1))∘Φ_loc (x).


Hence (<ref>) can be rewritten as 

    σ^(t_1,t_2)( Φ_loc (x))=σ^(t'_1+ε_1(1),t'_2+ε_2(1))( Φ_loc (x)).



On the other hand,  since σ^(t_1, t_2) (x) = σ^(t'_1, t'_2) (x'), we also have 

    σ^(t_1, t_2) (x) = σ^(t'_1+ε_1(1), t'_2+ε_2(1)) (x).


According to Lemma <ref>,   if ϕ is an affine equivalence then (<ref>) implies  (<ref>) and therefore (<ref>), as needed.
 
 The necessity of  the condition that  ϕ: B →B is an integer affine equivalence is obvious:   every fiberwise symplectomorphism induces an affine equivalence between B and B.









We now use Corollary <ref>  to prove Theorem <ref>. 

 First we apply Theorem <ref> which guarantees the existence of a real-analytic fiberwise diffeomorphism Φ_loc [In Theorem <ref>, this map was denoted by Φ.] between some neighborhoods of parabolic trajectories γ_0 ⊂ℒ_0 and γ_0⊂ℒ_0.  We now need to extend Φ_loc up to the desired fiberwise 
symplectomorphism Φ: U(ℒ_0)→U(ℒ_0).  

According to Corollary <ref> such an extension exists for all Liouville tori  (more precisely,  we only need to consider “wide” Liouville tori  because all “narrow” Liouville tori are already contained in the domain of Φ_loc).  Thus, it remains to explain why this map can be extended by continuity to each singular fiber.   

 On  Fig. 3 we can see the domain U on which Φ_loc is already defined and the complementary domain W to which Φ_loc should be extended.  Without loss of generality we may assume that  both domains are bounded by the sections 𝒩_1 and 𝒩_2.  Namely, U is located to the right of 𝒩_1 and 𝒩_2 and contains all singular orbits including the parabolic one.  The complimentary domain W is located to the left of 𝒩_1 and 𝒩_2 and contains no singularities at all.     
 
 Let y ∈ W be an arbitrary point located on one of singular fibers and V(y) be a sufficiently small neighborhood of y.   Then there exists (t_1, t_2)∈^2 such that σ^(-t_1, -t_2)(V(y)) ⊂ U and we may apply our extension formula (<ref>) to define Φ on V(y).    Obviously, this formula defines a real-analytic fiberwise (local) symplectomorphism from V(y) to its image in U(ℒ_0) and moreover, due to the uniqueness of such an extension, this map coincides with Φ that has been already defined on non-singular fibers (Liouville tori).   This is equivalent to saying that Φ can be naturally extended (by continuity) from Liouville tori to  all singular fibers. This completes the proof.  



  Our final remark is that the statement of Theorem <ref> given in Remark  <ref> can be also understood in terms of natural affine structures defined on B and B.   

A necessary and sufficient condition for the existence of a semi-local fiberwise symplectomorphism  Φ: U(ℒ_0) → U(ℒ_0) between neighborhoods of two cuspidal tori ℒ_0 and ℒ_0 is that the corresponding bases B and B are locally equivalent as manifolds with singular integer affine structures.  Moreover, every affine equivalence ϕ:B → B can be lifted up to a fiberwise symplectomorphism Φ. 

Thus, our paper gives a partial answer to Problem 27 from the collection <cit.> of  open problems in the theory of finite-dimensional integrable systems.  








§ APPENDIX




In this appendix we give a formal proof of the statement made in Remark <ref>,  namely we prove the following



Let P be a parabolic point of a momentum mapping ℱ=(F,H): M^4 →^2 in the sense of Definition <ref>  we   (in particular,  dF(P) 0) and 

    H =  H (H,F),    F =  F (H,F)
 
be a non-degenerate transformation such that d F(P)  0.   Then P is still parabolic w.r.t. H and F.







First of all, we notice that in Definition <ref>  we can replace H by H-const F and, in particular,   by H - kF  where k∈ is chosen in such a way that d(H-kF)=0.  In other words, without loss of generality we may assume that dH(P)=0   and similarly for H.  
Under this additional assumption, the quadratic differential d^2 H(P) makes sense on the whole tangent space T_PM^4. Taking into account that the tangent space to the hypersurface { F = F(P)} coincides with the kernel of the differential dℱ  (here we use the fact that dℱ (P) =  dF(P) = T_P{ F = F(P)}), we can reformute the first condition (i) as follows:

(i)  the rank of the restriction d^2 H (P)|_ dℱ equals 1.

The advantage of such a reformulation is that now this condition does not depend on the choice of F at all.   Therefore to verify the invariance of Condition (i) w.r.t. transformation (<ref>), it is sufficient to prove 



Let H =  H(H, F)  and  d H (P) = 0,  then the forms d^2 H (P)|_ dℱ and d^2  H (P)|_ dℱ are proportional with a non-zero factor.




It is sufficient to compare the Taylor expansions of  H and H̃  at the point P up to second order terms.   Let 

    Δ H ≃  a_1 Δ H + a_2 Δ F + a_11Δ H^2 + 2 a_12Δ H Δ F  +  a_22Δ F^2 …

Since d H(P) = dH(P) = 0, we conclude that a_2 = 0 and a_1 0 and obtain:

    Δ H ≃a_1/2 d^2 H (Δ x, Δ x) + a_22( dF(Δ x)) ^2 + …

(all the other terms are of order ≥ 3 and we omit them) or equivalently:

    d^2  H = a_1 d^2 H  + 2a_22  dF ⊗ dF

(the differentials and second differentials are taken at the point P). Restricting to dF =  dℱ,  we get d^2  H (P)|_ dℱ = a_1 d^2  H (P)|_ dℱ with a_1 0, as required.
 





Suppose  that Condition (i) holds.  Then the invariance of Condition (iii) w.r.t. transformation (<ref>)  amounts  to the following 



 Let H =  H(H, F), d H (P) = 0 and (d^2 H)|_ dℱ = 1.  Then the conditions  d^2 H (P) = 3 and d^2  H (P) = 3 are equivalent.



It is sufficient to use formula (<ref>), namely:

    d^2  H = a_1 d^2 H  + 2 a_22  dF ⊗ dF.
 
In general, these two forms d^2  H  and d^2 H do not necessarily have the same rank,  but under the condition that (d^2 H)|_ dℱ = 1   (using dℱ =  dF), the statements  d^2  H =3 and d^2  H =3  become equivalent (simple exercise in Matrix Algebra). Indeed,  if we choose a basis  e_1, e_2, e_3, e_4 in such a way that e_1, e_2, e_3 span dF and e_2, e_3 span d^2 H|_ dF and, in addition, dF(e_4)=1 we will see  that in matrix terms, the above formula (<ref>) can be rewritten as

    d^2 H = [ α 0 0 β; 0 0 0 γ; 0 0 0 δ; β γ δ λ ],       d^2  H = a_1  [ α 0 0 β; 0 0 0 γ; 0 0 0 δ; β γ δ λ ] + 2a_22[ 0 0 0 0; 0 0 0 0; 0 0 0 0; 0 0 0 1 ].

Now it is easy to see that both statements d^2 H = 3 and d^2  H = 3 are equivalent to the condition (γ,δ) (0,0), which completes the proof. 


Finally, we need to verify the invariance of Condition (ii) w.r.t. transformation (<ref>).
We first show that this condition does not change if we change F.       


Consider a new function  F =  F ( F , H).   Since scaling F↦const· F does not affect the surface { F = F(P)}, we may assume that  ∂ F/∂ F|_ℱ(P) = 1.    

According to the definition of v^3 H_0 we need to differentiate the same function H but along two different curves  γ(t) ⊂{ F = F(P)} and γ̃(t)⊂{ F =  F(P)}.


Let us choose local coordinates x_1, …, x_n on M in such a way that x_1 = F  and P=(0,…, 0). It is easy to see that if we set x̃_1 =  F, then still 
x̃_1, x_2, … , x_n is a good coordinate system. Moreover, the Jacobi matrix of the corresponding transformation at P is the identity.                                               


In coordinates x_1, …, x_n, the curve γ(t) can be defined as  γ(t) = (0, x_2(t), …, x_n(t)).  The curve γ̃(t) in the same coordinate system will be defined as 

    γ̃(t) = (x_1(t),  x_2(t), …, x_n(t)) = γ(t) + (x_1(t), 0, …, 0)

(all functions are the same except for  x_1(t) which should be chosen in such a way that  F = 0 along the curve.  In other words, x_1 can be found as a function of the other variables x_2,…, x_n from the implicit relation

    0= F ( F, H)=  F (x_1,  H(x_1, x_2,…, x_n))   ⇔    x_1=g (x_2,…, x_n)
  
and correspondingly x_1(t) = g(x_2(t), …, x_n(t)).


Our goal is to show that 

    d^3/dt^3 |_t=0 H (γ̃(t)) = d^3/dt^3 |_t=0 H (γ (t)).


It is easy to see that the Taylor expansion of  x_1=g (x_2,…, x_n) starts with quadratic terms.  Indeed, if   (w.l.o.g. we assume that  F(P)=0 and H(P)=0  so that Δ F = F and Δ H = H)

    F (F, H) =   F + b_2 H + b_11 F^2 + 2 b_12 FH + b_22 H^2 + …

then we need to resolve the equation (with respect to x_1)

    0= x_1 + b_2 H + b_11 x_1^2 + … =  x_1 + b_2  1/2∑_i,j=1^n ∂^2 H/∂ x_i∂ x_j x_ix_j + b_11 x_1^2 +…


These quadratic terms are sufficient to reconstruct the quadratic terms of the Taylor expansion of the function x_1=g(x_2,…, x_n).   This can be done by using the implicit function theorem, but we can also use the substitution

    x_1 = g(x_2, …, x_n) = ∑_i,j=2^n  c_ij x_i x_j + …

into (<ref>).  If we collect  (after substitution) all quadratic terms we obtain:

    0 = ∑_i,j=2^n  c_ij x_i x_j + b_2  1/2∑_i,j=2^n ∂^2 H/∂ x_i∂ x_j x_ix_j  + …

which means that up to a constant factor the quadratic expansions of g and H_0 coincide:

    g(x_2,…, x_n) = - b_2/2∑_i,j=2^n ∂^2 H/∂ x_i∂ x_j x_ix_j + …

or

    d^2 g (ξ, ξ) = - b_2 d^2 H (ξ, ξ)




Now we are ready to verify (<ref>). We have:

    d^3/dt^3 |_t=0 H (γ̃(t))  = 
    d^3H (γ̃', γ̃',γ̃') + 3d^2H (γ̃',γ̃”) + dH (γ̃”'),

and 

    d^3/dt^3 |_t=0 H (γ (t))  = 
    d^3H (γ', γ',γ') + 3d^2H(γ',γ”) + dH(γ”').



Since γ' = γ̃' = v and dH (P) = 0, we only need to compare the middle terms. In the second relation this term vanishes because  γ”∈ dF (as the first component of γ(t) identically vanishes) and γ' belongs to the kernel of (d^2H)|_ dF.


Consider the difference between d^2H (γ̃',γ̃”) and d^2H ( γ', γ”).  Since γ̃' = γ', we have 

    d^2H (γ̃',γ̃”) - d^2H ( γ', γ”) = d^2 H (γ', (γ̃- γ)”).

But γ̃- γ = (g(x_2(t), …, x_n(t)), 0, …, 0) so that for the potentially non-zero component of γ̃- γ we get

    d^2/dt^2|_t=0 g(x_2(t), …, x_n(t)) =  d^2 g (γ', γ') =  = - b_2   d^2 H (γ',γ').

It remains to notice that d^2 H (γ',γ') = 0 as v=γ' ∈ (d^2 H)|_ dℱ. Thus, (γ̃- γ)”=0.

Thus, we have shown that condition (iii) does not depend on the choice of F (keeping H fixed). The last step is to show that  (iii) does not depend on the choice of H  (keeping F fixed).  

Again we use the Taylor expansion (<ref>), but now up to third order terms

    Δ H = a_1 Δ H +  2 a_12Δ H Δ F + a_22 (Δ F)^2 + a_222(Δ F) + …,    a_1 0.

We do not need other terms  (like (Δ H)^2 for example)  as in local coordinates Δ H starts with quadratic terms.  This formula shows that under the condition Δ F = 0,  the Taylor expansions of H and H  (up to cubic terms) are proportional with a non-zero factor.  In particular for any curve γ(t)  lying on the surface { F = F(P)} = {Δ F = 0} we have 

    d^3/dt^3|_t=0 H(γ(t)) = a_1  d^3/dt^3|_t=0 H(γ(t))     
    v^3  H_0 = a_1  v^3  H_0,

as needed.   This shows that (ii)  is invariant under transformations (<ref>) completing the proof of Proposition <ref>.



We also want to explain one important phenomenon mentioned in the proof of Proposition  <ref>: for hyperbolic points, this coefficient in front of logarithm is known to be proportional to the period of the second (invisible in the real setting) cycle on the complex leaf ℒ_H,λ.  More rigorously, this statement can be formulated as follows.

Consider an analytic integrable system with one degree of freedom with the Hamiltonian of the form  H=xy and symplectic structure ω = f(x,y) dx∧ dy.  


Thinking of x and y as real variables, consider one-parameter family of curves  

    γ_H = { xy = H,  0<x≤ 1,  0<y ≤ 1}

and the function   (cf. Section <ref>)

    Π (H) = ∫_γ_Hω/dH  .




It can be easily checked by an explicit computation that this function has the following asymptotics at zero:

    Π(H) = a(H) ln H + b(H),

where a(H) and b(H) are both real-analytic in a neighborhood of zero. Similar to Section <ref>, here we integrate along a trajectory of the Hamiltonian flow between two sections N_1 = {y=1} and N_2 = {x=1}.    If we change these sections, then b(H) changes too whereas a(H) remains the same so that a(H) has an invariant meaning, i.e., does not depend on the choice of local coordinates (x,y). 


On the other hand if we think of x and y as complex variables, then the level { xy = H}, from the real viewpoint, is a surface locally homeomorphic to a cylinder  which contains a non-trivial cycle of the form

    γ̂_H ={ x(t) = H e^it,   y(t) = e^-it},
  
so that we can introduce another function 

    Π̂(H) = ∫_γ̂_Hω/dH.

This function is analytic in H.

The relation between Π (H) and Π̂(H) is given by the following



We have a(H) = ±1/2π iΠ̂(H)  or equivalently,

    Π (H) =  ±1/2π iΠ̂(H) ln H  + b(H)

with b(H) being analytic[The sign ± reflects the fact that both Π (H) and Π̂(H) depend on the choice of orientations on the curves γ_H and γ̂_H.].




One can proof this fact by using monodromy arguments  (which is nice and conceptual), but we will use a well known “isochore Morse lemma” <cit.>  which allows to get this result by an explicit computation:  there exists a local coordinate system such that

    H = xy         ω = f(H) dx∧ dy.


The form ωdH  can be replaced by the form f(H) y^-1 dy and we obtain:

    Π̂(H) = ∫_γ̂_Hω/dH = ∫_γ̂_H f(H) y^-1 dy = f(H) ∫_0^2π  ( e^-it )^-1 d ( e^-it ) =
    -2π  i  f(H)


On the other hand,   γ_H can be parametrised as  y=t,  x=H t^-1, t∈ [H, 1] and we get:

    Π (H) = ∫_γ_Hω/dH = ∫_γ_H  f(H) y^-1 dy = f(H) ∫_H^1 dt/t = - f(H) ln H.


Comparing these formulas for Π̂(H) and Π (H)  gives the required result.  


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