The Chazy XII Equation and Schwarz Triangle Functions
Oksana Bihun Sarbarish Chakravarty
math.CA math-ph math.MP nlin.SI








The Chazy XII Equation and Schwarz Triangle Functions

The Chazy XII Equation
 and Schwarz Triangle Functions

Oksana BIHUN and Sarbarish CHAKRAVARTY
O. Bihun and S. Chakravarty
Department of Mathematics, University of Colorado, Colorado Springs, CO 80918, USA
mailto:obihun@uccs.eduobihun@uccs.edu, mailto:schakrav@uccs.eduschakrav@uccs.edu

Received June 21, 2017, in final form December 12, 2017; Published online December 25, 2017

Dubrovin [Lecture Notes in Math., Vol. 1620, Springer, Berlin, 1996, 120–348]
showed that the Chazy XII equation y”'- 2yy”+3y'^2 = K(6y'-y^2)^2, K ∈ℂ,
is equivalent to a projective-invariant equation for an affine connection
on a one-dimensional complex manifold with projective structure.
By exploiting this geometric connection it is shown that the Chazy XII
solution, for certain values of K,
can be expressed as y=a_1w_1+a_2w_2+a_3w_3
where w_i solve the generalized Darboux–Halphen system. 
This relationship holds only for certain values
of the coefficients (a_1,a_2,a_3) and the Darboux–Halphen parameters
(α, β, γ), which are enumerated in Table <ref>.
Consequently, the Chazy XII solution y(z) is parametrized by a particular
class of Schwarz triangle functions S(α, β, γ; z)
which are used to represent the solutions w_i of the Darboux–Halphen
system. The paper only considers the case where α+β+γ<1.
The associated triangle functions are related among themselves
via rational maps that are derived from
the classical algebraic transformations of hypergeometric functions.
The Chazy XII equation is also shown to be equivalent to a Ramanujan-type
differential system for a triple (P̂, Q̂,R̂).

Chazy; Darboux–Halphen; Schwarz triangle functions; hypergeometric

34M45; 34M55; 33C05



§ INTRODUCTION

In 1911, J. Chazy <cit.> considered the classification problem of all third order differential equations of the form y”' = F(z, y, y', y”) possessing the Painlevé property, where the prime ' denotes d/ dz, and F is a polynomial in y, y', y” and locally analytic in z. A differential equation in the complex plane is said to have the Painlevé property if its general solution has no movable branch points. In his work, Chazy introduced thirteen classes of equations referred to as Chazy classes I–XIII. Among these, classes III and XII are particularly
interesting as their general solutions possess a movable barrier, i.e., a closed curve in the complex plane across which the solutions can not be analytically continued. That is, the solutions are analytic (or meromorphic) on either side of the barrier depending on the prescribed initial conditions. Both the Chazy III and XII equations can be expressed together as

    y”'- 2yy”+3y'^2 = K(6y'-y^2)^2,      K=0          K=4/36-k^2 .

The standard form for the Chazy XII equation is given by (<ref>) with the parameter K=4/(36-k^2), whereas the Chazy III equation
corresponds to K=0, and is the limiting case of Chazy XII
as k →∞. Chazy <cit.> observed the remarkable fact that (<ref>)
is linearizable via the hypergeometric equation

    s(s-1)χ”+ (7s6 -)χ'+(14k^2-1144)χ=0,

for k ∈ℂ∪∞, k≠ 0. He also showed that (<ref>)
possesses the Painlevé property when K=0 or when the value
of the parameter k is a positive integer such that k > 1
and k ≠ 6 (see also <cit.>). The k=0 case is linearizable
by Airy's equation χ”=csχ, c constant <cit.>, and the limiting cases
of k=± 6 can be solved via elliptic functions <cit.>.
For K=0 or for integer values k>6, the general solution of (<ref>) possesses a movable natural barrier in the complex plane.

The Chazy III equation arises in
mathematical physics in studies concerning magnetic mono­po­les <cit.>,
self-dual Yang–Mills and Einstein equations <cit.>, topological
field theory <cit.>, as well as special reductions of hydrodynamic type
equations <cit.> and incompressible fluids <cit.>.
The Chazy XII equation is related to the generalized
Darboux–Halphen system <cit.>, which arises
in reductions of self-dual Yang–Mills equations associated with
the gauge group of diffeomorphism Diff(S^3) of a 3-sphere <cit.>
as well as SU(2)-invariant hypercomplex manifolds <cit.>
with self-dual Weyl curvature <cit.>.
More recently, the Chazy XII equation with specific values of the parameter k
has been linked to the study of vanishing Cartan curvature invariant for
certain types of rank 2 distributions on a 5-manifold <cit.>.
Consequently, there has been renewed interest in the study of (<ref>) and
the singularity structure of its solutions in the complex plane. Interested
readers are referred to the comprehensive works by Bureau <cit.>,
Clarkson and Olver <cit.> and Cosgrove <cit.>.

A significant aspect of (<ref>) is the fact that its general solutions
can be expressed in terms of Schwarz triangle functions which
define conformal mappings from the upper
half complex plane onto a region bounded by three circular arcs
(see, e.g., <cit.>). For K=0,
solutions of the Chazy III equation are related to the automorphic forms
associated with the modular group SL_2(ℤ) and its subgroups <cit.>.
This fact can be traced back to the work of S. Ramanujan. In 1916,
Ramanujan <cit.>, <cit.>
introduced certain functions P(q), Q(q), R(q), q := e^2 π iz, Im(z)>0, which correspond to the
(first three) Eisenstein series associated with the modular group SL_2(ℤ).
He showed that these functions satisfy the differential relations

    1/2 π i P'(z) =P^2-Q/12,     1/2 π iQ'(z)=PQ-R/3,     1/2 π i R'(z) =PR-Q^2/2.

System (<ref>) can be reduced to a single third order differential
equation for P(q). In fact, the function y(z):= π i P(q) satisfies
the Chazy III equation.

For K ≠ 0, the solutions of the Chazy XII equation are also related
to Schwarz functions automorphic on curvilinear triangles that tessellate
the interior of the natural barrier
for integer values of the parameter k > 6. For 2 ≤ k ≤ 5
the solutions of (<ref>) are expressed via the
polyhedral functions which are rational functions
associated with symmetry groups of solids (see, e.g., <cit.>).

In this paper, we primarily consider the Chazy XII equation (<ref>)
for integer values k > 6. The main objective of this paper
is to find all solutions y(z) that are given in terms of Schwarz
triangle functions. The central result to achieve this goal is
the following:

If y(z) = 6Y(z) solves (<ref>) with K ≠ 0, then
Y(z) is a convex linear combination

    Y(z) = β_1w_1 + α_1w_2 + γ_1w_3,     α_1, β_1, γ_1 > 0,     α_1+β_1+γ_1=1

of the variables w_1, w_2, w_3 satisfying the Darboux–Halphen system (<ref>) with
parameters (α,β,γ). The relationship holds only
for special choices of (α_1, β_1) and (α, β, γ)
listed in Table <ref> of Section <ref>. The variables w_1, w_2, w_3
are expressed in terms of the Schwarz triangle function s(z) = S(α, β, γ; z) and its derivatives as shown in (<ref>).

The above result is obtained by extending the geometric formulation
of (<ref>) by B. Dub­ro­vin <cit.> to recast (<ref>) as
a purely algebraic relation. The Chazy XII solution y(z) is expressed
in a parametric form involving hypergeometric functions _2F_1

    y = y(s),      z=z(s) ,

where s(z) = S(α, β, γ; z) is the Schwarz triangle function
(see Section  <ref> for details).
Thus, Table <ref> provides explicit parametrizations
of the Chazy XII solutions in terms of a distinct family of Schwarz
triangle functions. This family includes previously known functions, e.g., S(,13,1k;z), 6 < k ∈ℕ as well as
some new cases. We believe that
our approach is new, and that it can be applied to study similar
nonlinear equations with natural barrier <cit.>.

The paper is organized as follows. Section <ref> develops necessary
geometric background on affine connections on a one-dimensional
complex manifold with a projective structure fol­lowing <cit.>.
In Section <ref>, the generalized Darboux–Halphen system is introduced and its
solutions in terms of Schwarz triangle functions are discussed. The
connection between the Chazy XII equation and the
generalized Darboux–Halphen
system described in Section <ref> leads to a purely algebraic formulation
of (<ref>). The solvability conditions of this algebraic system
are analyzed in Section <ref>, and these lead to a classification of the
Chazy XII solutions according to their pole structures inside
the natural barrier.
In Section <ref>, a Ramanujan-type triple of functions is introduced.
These functions satisfy a system of first order equations
that is similar to (<ref>) and is equivalent to the Chazy XII equation.
In Section <ref>, the Schwarz triangle function is presented as
the inverse to the conformal map defined by the ratio of two linearly
independent solutions of the hypergeometric equation. The parametrizations
of the Ramanujan-like triple as well as the Chazy XII solution in terms
of hypergeometric functions are also discussed here. Section <ref> outlines
the rational transformations between the distinct Schwarz
functions parameterizing the Chazy XII solution. These rational maps are
derived from the well-known algebraic transformations among the
hypergeometric functions. In order to make the paper self-contained,
we have included two appendices. Appendix <ref> contains a proof of Lemma <ref> introduced
in Section <ref>, while Appendix <ref> contains an elementary derivation of the
radius of the natural barrier for the Chazy XII solution.



§ BACKGROUND



 §.§ Affine connection and projective structure

We begin this section by reviewing the relation between the solution y(z)
in (<ref>) with affine connections on a one-dimensional
complex manifold.
We consider differential forms of order m ∈ℕ denoted by
f=f(z) dz^m on a one-dimensional
complex manifold with local coordinate z, where f(z) is a holomorphic
(or meromorphic) function. Under the local change of coordinates z →z̃(z),
f transforms according to f(z)dz^m = f̃(z̃) dz̃^m. The covariant
derivative of a m-differential is a (m+1)-differential ∇ f
defined by

    ∇ f = ∇ f(z)  dz^m+1,     ∇ f(z):= f'(z) - mη(z)f(z) ,

where η = η(z) dz is a holomorphic (or meromorphic) affine connection on the manifold. It follows from the transformation property of ∇ f that η must transform as

    η̃(z̃) dz̃ = η(z)  dz -
    (z̃”(z)/z̃'(z)) dz .

The “curvature” associated with η is defined by the quadratic differential Ω= Ω(z)dz^2 as

    Ω(z):= η'(z) - η(z)^2 .

In fact, Ω is a projective connection transforming under local
change of coordinates as

    Ω̃(z̃) dz̃^2 =
    Ω(z) dz^2 - {z̃;z} dz^2 ,     {z̃, z} := z̃”'(z)/z̃'(z)-3/2(z̃”(z)/z̃'(z))^2 ,

where {z̃, z} is called the Schwarzian derivative.
Under the projective (Möbius) transformations

    z →z̃ = az+b/cz+d ,      ad-bc ≠ 0 ,

Ω transforms covariantly, i.e.,
Ω(z) dz^2 = Ω̃(z̃) dz̃^2
because {z̃, z}=0. Consequently, any m-differential of the form
Q = Q(z) dz^m, where Q(Ω(z), ∇Ω(z), ∇^2Ω(z), …)
is a homogeneous polynomial of degree m with ∇^rΩ = r+2,
transforms covariantly and an equation of the form Q=0 remains invariant under a projective transformation.
It was shown in <cit.> that equation (<ref>)
corresponds to the projective-invariant equation Q=0 for m=2, i.e.,

    ∇^2Ω + JΩ^2=0

for any constant J. This yields
the following third order differential equation for η(z):

    η”' - 6ηη” + 9η'^2 +(J-12)(η'-η^2)^2 =0 ,

which reduces to (<ref>) by setting y=3η and 108K=(12-J). Thus, the Chazy III and Chazy XII equations (<ref>) can be interpreted as a certain differential polynomial
invariant of an affine connection in a one-dimensional complex manifold with a projective structure.

A projective structure on a one-dimensional complex manifold
M ⊆ℂℙ^1 is defined by an atlas
of local coordinates with transition functions given by Möbius transformations.
In a local coordinate chart, the Möbius transformations are generated by the
vector fields ⟨∂_z, z∂_z, z^2∂_z ⟩ isomorphic
to the Lie algebra 𝔰𝔩_2(ℂ). A nontrivial
representation of 𝔰𝔩_2(ℂ) is given by the
vector fields ⟨ u_1^2∂_s, u_1u_2∂_s, u_2^2∂_s⟩
where u_1(s), u_2(s) is a pair of linearly
independent solutions of the complex Schrödinger equation

    u”+14 V(s)u=0 ,

(where the factor 14 is inserted for convenience). Then the identification
of the vector field ∂_z with u_1^2∂_s induces a local change
of coordinates s → z(s) on M via the ratio

    z(s) = u_2(s)/u_1(s) .

If M is not simply connected, the monodromy group G ⊂ GL_2(ℂ)
resulting from the analytic extensions of the pair
(u_2, u_1) → (au_2+bu_1, cu_2+du_1) along all
possible closed loops in M acts projectively
on the ratio in (<ref>) via the Möbius transformations

    z →γ(z) := az+b/cz+d ,     [ a b; c d ]∈ G.

The projectivized monodromy group
is the quotient group Γ≅ G/λ I_2 ⊆ PSL_2(ℂ),
where I_2 is the 2 × 2 identity matrix. A different choice for
the basis (u_1,u_2) would lead to a different projective structure on M.

Note that the Schwarzian derivative is a differential invariant of the projective transformation, i.e., {γ(z);s}={z;s}. For a projective structure induced by the Schrödinger equation (<ref>), z(s) satisfies the third order Schwarzian equation


    {z;s} =  V(s) ,

which can be linearized via (<ref>) and
(<ref>). It follows that the inverse function s(z) (if it exists
globally) is a projective invariant function on the manifold M with the
automorphism s(z) = s(γ(z)), γ∈Γ, and satisfies

    {s;z} +  V(s) s'(z)^2 =0 .


An affine connection on the one-dimensional complex manifold M can be defined
uniquely by its trivialization η(s)=0 in the projective invariant coordinate s.
Then the transformation
s → z(s)=u_2(s)/u_1(s) in terms of the solutions of the Schrödinger equation
leads to the following expression of the affine connection in the z-coordinate

    η(z) = s”(z)/s'(z)= - z”(s)/z'(s)^2 = 2 u_1(s)u_1'(s) ,

using the transformation rule (<ref>). Furthermore, from
(<ref>), the differential Ω(z) is
then given in terms of the Schrödinger potential V(s) as

    Ω(z) = {s;z}= -{z;s}/z'(s)^2 = - u_1^4(s) V(s) .

Consequently, the solutions of the Chazy III and XII equations (<ref>),
which are equivalent to (<ref>), can be expressed via
the solutions and the potential of the complex Schrödinger equation (<ref>). It is worth noting that if the affine connection is trivialized
in a different coordinate x(s) with dx = f(s) ds, then
from (<ref>), η(z) dz =  dlog(f(s)s'(z)).
We will utilize this fact and the geometric framework discussed above to construct a solution method in Section <ref> for (<ref>), by exploiting its relationship to another nonlinear differential system described below.



 §.§ The Halphen system and Schwarz triangle functions

In 1881, Halphen considered a slight variant <cit.>
of the following nonlinear differential system

    w_1'=-w_2w_3+w_1(w_2+w_3)+τ^2, 
    
    w_2'=-w_3w_1+w_2(w_3+w_1)+τ^2, 
    
    w_3'=-w_1w_2+w_3(w_1+w_2)+τ^2, 
    τ^2 = α^2(w_1-w_2)(w_2-w_3)+β^2(w_2-w_1)(w_1-w_3)+
    γ^2(w_3-w_1)(w_2-w_3) ,

for functions w_i(z) ≠ w_j(z), i≠ j, where i,j ∈{1,2,3}, and
α, β, γ are constants.
A special case of this equation with α = β = γ = 0 originally
appeared in Darboux's work of triply orthogonal surfaces on ℝ^3 in
1878 <cit.>. Its solution was given by Halphen <cit.>
in 1881 in terms of hypergeometric functions. Subsequently, Chazy <cit.>
showed that y(z):=2(w_1+w_2+w_3) satisfies the Chazy III equation
introduced in Section <ref>.
More recently, (<ref>) was re-discovered as a certain symmetry
reduction of the self-dual Yang–Mills equations, and was referred to as the
generalized Darboux–Halphen (gDH) system <cit.>.

The gDH system can be solved via the Schrödinger equation (<ref>) with the potential

    V(s)=1-α^2/s^2+1-β^2/(s-1)^2+ α^2+β^2-γ^2-1/s(s-1) ,

and defining a projective structure on M via the ratio z(s) in (<ref>) as described in Section <ref>.
Note that in this case (<ref>) is a second order Fuchsian differential
equation with three regular singular points, and
α, β, γ are the exponent differences (for any
pair of linearly independent solutions u_1 and u_2) prescribed at the
singular points 0, 1 and ∞, respectively. The generators of the
projectivized monodromy group Γ are determined by the exponent
differences. If the gDH variables w_1(z), w_2(z), w_3(z)
are expressed in terms of the projective invariant inverse
function s(z) (and its derivatives) as follows:

    w_1 = 1/2[log(s'/s)]',   
    w_2 = 1/2[log(s'/s-1)]',   
    w_3 = 1/2[log(s'/s(s-1))]',

then a straightforward calculation shows that (<ref>) reduces
to the Schwarzian equation (<ref>) for s(z), where the constants
α, β, γ in V(s) are the
same as those appearing in τ^2 of (<ref>).
Equation (<ref>) is equivalent to (<ref>) (after interchanging
the dependent and independent variables) which is then reduced to (<ref>).
Thus, the gDH system can be effectively linearized
by the Schrödinger equation (<ref>) with the potential V(s) given by (<ref>).

The ratio z(s) in (<ref>) of
any two linearly independent solutions u_1, u_2 of (<ref>)
with the potential V(s) given by (<ref>) defines a conformal mapping
that was studied extensively by H.A. Schwarz <cit.> in 1873.
The map z(s) is, in general, branched at the regular singular
points s=0, 1, ∞.
However, if the parameters α, β, γ are either
zero or reciprocals of positive integers, and satisfy
α+β+γ < 1, then the mapping z(s) defines a plane
region D, which is tessellated by an infinite number of
non-overlapping hyperbolic, circular triangles on the complex z-plane.
The interior of each triangle is an image of the upper (lower)-half
s-plane under the map z(s) and its analytic extensions, and is
bounded by three circular arcs forming interior angles απ, βπ, and γπ at the vertices z(0), z(1),
and z(∞), respectively. Two adjacent triangles are obtained via Schwarz reflection
principle, and are images of each other under reflection across
the circular arc that forms their common boundary.
In this case, the inverse s(z) is a single-valued, meromorphic, automorphic function whose automorphism
group is the projective monodromy group Γ associated with (<ref>) and (<ref>). That is, s(γ(z)) = s(z)
for all γ∈Γ where γ(z) is the Möbius transformation
defined in (<ref>).
When the exponent differences satisfy the conditions prescribed
above, Γ is a discrete subgroup of PSL(2,ℝ), and turns out to be
the group of Möbius transformations generated by an
even number of reflections across the boundaries of the circular triangles.
This automorphic inverse function

    s(z) := S(α, β, γ; z)

is called the Schwarz triangle function, and the automorphism group Γ is referred to as the triangle group.
It is worth noting that if α, β, γ in (<ref>) are either zero or reciprocals of positive integers,
but either α+β+γ =1 or α+β+γ >1,
then the map z(s) in (<ref>) tiles the z-plane
either into infinitely many plane triangles, or the extended z-plane
(Riemann sphere) into a finite number of spherical triangles, respectively.
The triangle group Γ is a simply or doubly periodic
group when α+β+γ =1, and corresponds to one of the
four symmetry groups of the regular solids when α+β+γ >1.
A detailed discussion of the automorphic groups can be found in the monograph <cit.>.

It follows from (<ref>) or from (<ref>) with
V(s) as in (<ref>) that the only possible singularities of s(z)
and its derivatives on the domain D are located at the vertices
of each triangle where s(z) takes the value of 0, 1, or ∞.
The boundary of D in the z-plane is a Γ-invariant circle C which
is orthogonal to all three sides of each triangle and its
reflected images. This orthogonal circle C is the set of limit points
for the automorphic group Γ, and corresponds to a dense set
of essential singularities which form a natural barrier for the
function s(z). In its domain of existence D, the only possible
singularities of s(z) are poles which correspond to the vertices
where s(z) = ∞.

In summary, when the parameters α, β, γ
in (<ref>) are either zero
or reciprocals of positive integers and α+β+γ <1,
the general solution of (<ref>)
is obtained as the unique inverse of the ratio

    z(s) = a u_2(s)+b u_1(s)/cu_2(s) +du_1(s),     
    a,b,c,d ∈ℂ,      ad-bc =1,

where u_1 and u_2 are two linearly independent solutions of (<ref>).
The solution is single-valued and meromorphic inside a disk in the extended
z-plane, and can not be continued analytically across the boundary of
the disk. This boundary is movable as its center and radius are
completely determined by the initial conditions, which depend
on the complex parameters a, b, c, d.
Recently, a number of new nonlinear differential equations
whose solutions possess movable natural boundaries have been
found <cit.>.
These can be solved by first transforming them
into a Schwarzian equation (<ref>) and then following the
linearization scheme described above.



§ CHAZY XII AND TRIANGLE FUNCTIONS

In this section we present a solution method for the Chazy XII equation
based on the geometric approach discussed in Section <ref>. Specifically, we exploit
the fact that the Chazy XII equation in (<ref>) is equivalent to
the coordinate independent form in (<ref>)
for the quadratic differential Ω. We express the affine connection η associated with Ω
via the gDH variables introduced in Section <ref> so that the Chazy XII
equation can be expressed in terms of the Schrödinger potential V(s)
in (<ref>) and its derivatives, in a simple algebraic fashion. Then the
solution of the Chazy XII equation is obtained via the
Schwarz triangle function S(α, β, γ; z) for
specific values of the triple (α, β, γ), which we identify.
Recall that in this article we only consider the Chazy XII equation.
The parametrization of the Chazy III equation by Schwarz
triangle functions was studied in <cit.>.



 §.§ The gDH system and the Chazy XII equation

Note first from (<ref>) and (<ref>) that
2w_i= dlog(f_i(s)s'(z))/ dz transforms as an affine connection
that is trivialized in the coordinate x_i(s) with dx_i=f_i(s) ds for

    f_1(s)= 1/s,      f_2(s)=1/s-1,     
    f_3(s)=1/s(s-1) .

In particular, under a Möbius transformation z →γ(z)
given by (<ref>) with ad-bc=1, w_i transforms as

    w_i(γ(z)) = (cz+d)^2w_i(z)+ c(cz+d),     γ∈Γ ,

which leaves the gDH system (<ref>) invariant.
Next let us define the function

    y(z) := a_1w_1 + a_2w_2 + a_3w_3,

in terms of the gDH variables w_i, where the coefficients a_i
are nonnegative constants.
Then from (<ref>), y(z) can be expressed in terms of s(z) and its derivatives as follows:

    y(z) = p/2 ϕ'(z)/ϕ(z),       ϕ(z) = s'(z)/s(z)^1-α_1(s(z)-1)^1-β_1,

with p:= a_1+a_2+a_3, α_1:= a_2/p and
β_1:= a_1/p and 0 ≤α_1, β_1 ≤ 1.
If we set p=6 above, then it is possible to define an affine connection
η such that in the z-coordinate,

    η(z)=y(z)/3= dlog(ϕ)/ dz.

It follows from (<ref>) that η is trivialized in
the coordinate x such that dx = ϕ(z) dz = f(s) ds
where f(s)=s^-1+α_1(s-1)^-1+β_1. Moreover,
in the s-coordinate,

    η(s)= (log f)'(s) = -1-α_1/s - 1-β_1/s-1 ,
        Ω(s) = η'(s)-η^2(s) ,

which are rational functions
of s. The curvature in the z-coordinate is obtained using the
transformation property (<ref>) as

    Ω(z) = (Ω(s) - {z;s})s'(z)^2 =
    (Ω(s) -  V(s))s'(z)^2 := -V_2(s)s'(z)^2,

where the second equality above follows from (<ref>) with
V(s) as in (<ref>). Then the covariant derivatives of the
quadratic differential Ω are given by the transformation rule

    ∇^mΩ(z) = -(∇^m V_2(s))s'(z)^m+2
    := -V_m+2(s)s'(z)^m+2 ,      m ≥ 1 ,

where V_m+2(s) are rational functions of s defined recursively
for n ≥ 2 as

    V_n+1(s) = ∇ V_n(s) = V_n'(s) - nη(s)V_n(s),
         V_2(s) =  V(s) - η'(s) + η(s)^2 ,

with η(s) defined in (<ref>).

Our next goal is to derive conditions under which Ω(z) will
satisfy the projective-invariant equation (<ref>), equivalently,
the affine connection η(z) will satisfy (<ref>).
Then the function y(z)=3η(z) defined in (<ref>) with p=6 will
solve the Chazy XII equation in (<ref>). Henceforth, the
value p=6 will be used throughout the rest of this article.

It follows from the expressions for Ω(z) and
its covariant derivatives obtained above, that for m=2,
(<ref>) implies a simple
algebraic relation between rational functions V_2(s)
and V_4(s), namely,

    V_4(s) = J V_2^2(s),      J ≠ 12 ,

that should hold for all s. This condition imposes certain restrictions
on the parameters (α, β, γ) and (α_1, β_1)
appearing in the functions V_2(s) and V_4(s). It is worth pointing
out here that (<ref>) together with (<ref>)
lead to the central result advertised in Section <ref>.

In what follows, we
will systematically determine the sets of parameters for which
(<ref>) holds. In particular, we will identify the values of
the triple (α, β, γ) in the Schwarz triangle
functions S(α, β, γ; z) which determine the solution
of the Chazy XII equation y(z) via (<ref>).



 §.§ The Schwarz function parametrization

From equations (<ref>), (<ref>) and (<ref>),
the rational function V_2(s) can be written as

    V_2(s) = 1/2[A/s^2+B/(s-1)^2+C/s(s-1)] ,
    
    
    A= α_1^2-α^2,      B= β_1^2-β^2,     
    C=γ_1^2-A-B-γ^2,

where the parameters (α_1,β_1,γ_1) are defined as follows:

    α_1=a_2/6,     β_1=a_1/6 ,
        γ_1 = a_3/6,      0 ≤α_1, β_1, γ_1 ≤1,
        α_1+β_1+γ_1=1 .

Then V_4(s) is readily computed from the
recurrence relation in (<ref>). Upon substituting the expressions for V_2(s)
and V_4(s) into (<ref>) and rationalizing the resulting expression, one
finds that (<ref>) is satisfied if and only if

    u_1s^4+u_2s^3(s-1)+u_3s^2(s-1)^2+u_4s(s-1)^3+u_5(s-1)^4=0

for all values of s. The last identity is equivalent to the vanishing
of the coefficients u_i, i.e.,

    u_1 := JB^2-12Bβ_1^2 =0, 
    
    u_2 := 2JBC-4[(1-α_1)(1-6β_1)B +
     (1-2β_1)(1-3β_1)C] =0 , 
    
    u_3 :=J(2AB+C^2) -4[(2-3β_1)(1-β_1)A+(2-3α_1)(1-α_1)B
    
     + [(2-3β_1)(1-2α_1)+(2-3α_1)(1-2β_1)]C] =0 ,
    
    
    u_4 :=2JAC-4[(1-β_1)(1-6α_1)A+(1-2α_1)(1-3α_1)C]=0 ,
    
    
    u_5 := JA^2-12Aα_1^2=0 .



System (<ref>) represents a set of coupled, algebraic equations for
the parameters (α,β,γ), (α_1, β_1) and J
subject to the conditions

    α, β, γ∈{1n ,   n ∈ℕ}∪{0},
        α+β+γ<1,      0 ≤α_1, β_1 ≤ 1,

and J ≠ 12. The case J=0 corresponds
to K=19 or k=0 in (<ref>). The Chazy XII equation for this
case can be linearized via Airy's equation as mentioned in
Section <ref>. Hence, the J=0 case is distinct from
the cases considered here which correspond to Schrödinger potential
V(s) in (<ref>) with three regular singular points. Furthermore,
the following lemma is proven in Appendix <ref>.
 
If J=0, then (<ref>) has no admissible solution satisfying conditions
(<ref>).

Henceforth, we take J ≠ 0 through the remainder of the main text of
this paper.
 
If a triple (α, β, γ) of non-negative numbers
satisfying α+β+γ < 1
solves the system (<ref>) with J ≠ 12,
then α≠ 0, β≠ 0, and γ≠ 0.


If α=0, then A = α_1^2 from (<ref>). The condition (<ref>)
with J ≠ 12 implies α_1=0. Hence, A=0. Then equations
(<ref>) and (<ref>) imply B=0 and C=0.
Consequently, β=β_1, γ=γ_1 from (<ref>).
Therefore, β+γ = β_1+γ_1 =1 contradicting
α+β+γ < 1.

If β=0, the proof is similar to above, starting from condition
(<ref>) first, then using equations (<ref>) and (<ref>).

If γ=0 then D:=A+B+C = γ_1^2. The condition
∑_i=1^5u_i=0 in (<ref>) yields
JD^2=12γ_1^2D, which implies that γ_1=0. Hence,
α_1+β_1=1 and D=0. Using the last condition to eliminate C from (<ref>) and (<ref>), one obtains


    JAB = (1-2β_1)(1-3β_1)A-(1-6β_1)(1-2α_1-β_1)B ,
    
    
    JAB = -(1-6α_1)(1-2β_1-α_1)A + (1-2α_1)(1-3α_1)B ,


where (<ref>) and (<ref>) are also used to eliminate terms
involving A^2 and B^2. It follows from (<ref>) and (<ref>) together
with α_1+β_1=1 that if either A=0,
or B=0, then both A and B vanish. Hence, α+β =
α_1+β_1=1, which contradicts α+β+γ < 1.

Finally, let A ≠ 0 and B≠ 0. Then
(<ref>) and (<ref>) imply that
JA=12α_1^2, JB=12 β_1^2.
Substituting these into (<ref>) and (<ref>) to
eliminate A and B, yields two equations
for α_1 and β_1. The resulting equations have no solution
(α_1,β_1) that satisfies the condition α_1+β_1=1.
Thus, γ can not be 0.

One can also make the inequalities for α_1, β_1, γ_1
in (<ref>) strict, i.e., 0 < α_1, β_1, γ_1 < 1.
 
If α_1, β_1, γ_1 satisfy (<ref>) and
solve the system (<ref>) with (α,β,γ) as in Lemma <ref>,
then α_1, β_1, γ_1 ∉{0,1}.


If α_1=0, then it follows from (<ref>) that A=0.
Then, (<ref>) implies that α = α_1=0, contradicting Lemma <ref>.
Thus α_1 ≠ 0. A similar argument starting from (<ref>)
shows that β_1 ≠ 0.

If α_1 =1, then (<ref>) implies that β_1=0, which is
not possible. Similarly, β_1=1 or γ_1=1 is not possible.

Finally, assume that γ_1 = 0. Since JD^2 = 12γ_1^2D
where D:= A+B+C (see Lemma <ref>), it follows that D=0.
Then, (<ref>) implies that γ = γ_1=0,
contradicting Lemma <ref>.


Lemma <ref> implies that
α, β, γ∈{1n,   n ∈ℕ}
in (<ref>).
Let s(z)=S(α,β,γ;z) be the triangle function
associated with those parameters, and let z_0 be
a vertex of a triangle in the domain of existence D of s(z)
such that s(z_0) := s_0 ∈{0,1,∞}. The map z(s) defined
via the equa­tions (<ref>), (<ref>) and (<ref>), behaves
locally near s=s_0 as (see, e.g., <cit.>)

    z(s)=z_0+(s-s_0)^μψ_1(s),      s_0 ∈{0,1}    
    
     z(s)=z_0+s^-μψ_2(s^-1),      s_0=∞ ,

where ψ_i is analytic near s_0 with ψ_i(s_0) ≠ 0, and
μ∈{α, β, γ} is the corresponding
exponent difference at the singular point s_0. Consequently, the inverse
s(z) is single-valued function defined locally as

    s(z) = s_0 + (z-z_0)^mϕ_1(z),      z_0 ∈{z(0), z(1)}    
     s(z) = (z-z_0)^-mϕ_2(z) ,      z_0=z(∞) ,

where ϕ_i(z) is analytic in the neighborhood of z=z_0 with
ϕ_i(z_0) ≠ 0, and m=μ^-1∈ℕ.
Thus, s-s_0 has a zero of order 1α, 1β at the vertices
z_0=z(0), z(1), respectively, and s(z) has a pole of order
1γ at the vertex z_0=z(∞) of each triangle in the domain D.
As mentioned earlier in Section <ref>, it is sufficient to examine the behavior
of the function s(z) and its derivatives near the vertices z(0), z(1),
and z(∞) of just one triangle inside the domain D. The Schwarz
reflection principle and the automorphic property then ensure
that s(z) will have the same behavior at the vertices of the
reflected triangles in D.
A straightforward calculation using the above behavior of s(z) in (<ref>) shows that y(z) is a meromorphic function in D with simple
poles at the vertices z(0), z(1), z(∞) of each triangle in D with
the residues

    Res_z(0)=3(α_1α-1),
        Res_z(1)=3(β_1β-1),
        Res_z(∞)=3(γ_1γ-1).

Of course, at the boundary of the domain D, y(z) inherits the same natural
barrier of essential singularities as the function s(z). Note that it is
possible for y(z) to be analytic at a vertex z_0 in the interior of D
provided that the residue vanishes at z_0, i.e., if either α=α_1,
β=β_1, or γ=γ_1. However, it is impossible for all three
residues to vanish simultaneously because then
α+β+γ = α_1+β_1+γ_1 =1, which violates the
condition α+β+γ < 1 in (<ref>).
The discussion above concerning the singularity structure of y(z)
in the interior of its domain of definition D
can be summarized in the following proposition.
 
The solution y(z) of the Chazy XII equation given by (<ref>) in terms of the triangle function
s(z)=S(α, β, γ; z) where α, β, γ
satisfy (<ref>), is meromorphic with only simple poles
inside its domain of definition D. These poles can
only occur at the vertices z(0), z(1) and z(∞) of the
circular triangles tessellating the domain D. Moreover, y(z)
must have at least one simple pole at one of the vertices of each triangle.

Hence, there are three distinct cases resulting from Proposition <ref>, namely, y(z) has a simple pole at
(1)  only one vertex z_0 ∈{z(0),z(1),z(∞)},
(2)  only two of the three vertices, and (3)  all three vertices.
The admissible parameters that satisfy (<ref>)
can be determined by considering each case separately.

Case 1. Suppose y(z) is analytic at z(0) and z(1), and
has a simple pole only at z(∞) on each triangle in D. Then the
vanishing of residues at z(0) and z(1) is equivalent to α=α_1
and β=β_1, which implies that A=B=0. Note however that C≠ 0
since Res_z(∞)≠ 0. In this case (<ref>) and (<ref>) are identically satisfied, while (<ref>) and (<ref>) imply
that (1-2α_1)(1-3α_1)=(1-2β_1)(1-3β_1)=0.
Hence,

    (α, β) ∈{(12,12), (12,13), (13,12)
    (13, 13)}.

Since the subcase α=β=12 is not admissible due to
the condition α+β+γ<1, and the subcases
(α, β)∈{(12,13), (13,12)}
are the same modulo a vertex permutation, there are only two distinct subcases,
namely, (α, β)=(12,13) and (α, β)=(13,13).

If (α, β)=(12,13), the remaining condition (<ref>)
yields JC=13. Choosing γ=1k, k∈ℕ,
gives J=12k^2/(k^2-36), which is consistent with the value of K
in (<ref>). Recall (from the definition below (<ref>))
that K=(12-J)/108.

If (α, β)=(13,13), then (<ref>) implies that
JC=43. In this case, choosing γ=1k, k∈ℕ
gives J=12k^2/(k^2-9) so that K=1/(9-k^2), which is different from
the value used by Chazy in (<ref>). However, the rescaling
k →k2 restores the value of K in (<ref>) while modifying
the value of γ to γ=2k.
If the simple pole of y(z) is chosen to be at z(0) or z(1) instead,
the values of the parameters are simply permuted. Thus, modulo permutations,
there are two possible sets of admissible parameters:

    (i)          (α, β, γ) = (12, 13, 1k),
    (α_1, β_1, γ_1) = (12, 13, 16),     

    (ii)         (α, β, γ) = (13, 13, 2k),
    (α_1, β_1, γ_1) = (13, 13, 13) ,

for the same value of K as in (<ref>), and the rescaled
(k → 2k) version of subcase (ii) above.

Case 2. Suppose now y(z) is analytic at only one vertex z(0) and
has a simple pole at each of z(1) and z(∞). Then α=α_1,
which means A=0, so that (<ref>) is automatically satisfied.
Here, B ≠ 0 and B+C ≠ 0 since the residues at z(1) and
z(∞) do not vanish. Then (<ref>) leads to three
subcases: (i) α_1=12, (ii) α_1=13, and
(iii) C=0.

In the case where α_1=, one solves first for β_1, then for B
and B+C from the remaining equations in (<ref>). The admissible
values of the parameters up to a permutation, are β_1=16, and
(α, β, γ) = (12, 1k, 2k), k∈ℕ
with the value of K as in (<ref>). However, one also obtains
(α, β, γ) = (12, 12k, 1k) with K=1/(9-k^2)
as in Case 1 above, which can be rescaled via k →k2 to
Chazy's choice for K.

If α_1=13, a similar analysis as above yields β_1=16,
and (α, β, γ) = (13, 1k, 3k), k ∈ℕ
with K as in (<ref>). There is also a second case with
(α, β, γ) = (13, 13k, 1k) and K=4/(36-9k^2),
which can be rescaled back to the previous case via k →k3.

Finally, if C=0, (<ref>) reduces to (2-3α_1)(1-α_1)B=0. Therefore, α_1=23 since α_1 = α≠ 1 and B≠ 0.
The remaining equations in (<ref>) yield β_1=16, β=γ=1k, k∈ℕ with K as in (<ref>).

Case 3. If y(z) has a simple pole at each of the vertices z(0), z(1)
and z(∞), then A ≠ 0, B ≠ 0, and A+B+C ≠ 0. The equations (<ref>) and (<ref>) imply, respectively, that JB=12β_1^2 and JA=12α_1^2. Substituting these into (<ref>) and (<ref>), one obtains

    (1-6β_1)[2(1-α_1)B+(1+β_1)C]=0 , 
    
    (1-6α_1)[2(1-β_1)A+(1+α_1)C]=0 .

The above equations lead to four distinct subcases:
(i) α_1=β_1=16, (ii) α_1=16, β_1 ≠16,
(iii) α_1 ≠16, β_1=16, (iv) α_1 ≠16,
β_1 ≠16. From the remaining condition (<ref>) and
the above equations one finds that the first three subcases yield, modulo
permutations, only one distinct set of admissible parameter values given by

    (α, β, γ)= (1k, 1k, 4k),      k∈ℕ,
         (α_1, β_1, γ_1)= (16, 16, 23) ,

with K as in (<ref>). Like the previous two cases, an additional
set (α,β,γ)= (14k, 14k, 1k)
with K=1/(9-4k^2) is also found by first choosing γ=1k and
then solving for α, β. This case can be reduced to the
one displayed above by the rescaling k → k4.

=1 Subcase (iv) leads to α_1=β_1 and α=β. Using
condition (<ref>), one obtains the equilateral triangle
corresponding to α=β=γ=2k, k∈ℕ
and α_1=β_1=γ_1=13 with K as in (<ref>).
It is also possible to choose α=β=γ=1k for the
same values of α_1 and β_1 but with K=1/(9-k^2). These
are related by the rescaling k → k2.

The results of our classification of the Chazy XII solution in terms of the
Schwarz triangle functions S(α,β,γ;z) are summarized
in Table <ref>. The parameters (α,β,γ) of the
triangle function are listed in the second column, while the parameters
(α_1,β_1) in the third column determine the corresponding
solution y(z) given by (<ref>). This solution has a simple pole
at one or more of the vertices z(0), z(1), z(∞) as stated in
Proposition <ref>. The residue at each simple pole is listed in the
last column. The values of the Chazy parameter K are given in the
fourth column. The parameter values listed in each row are modulo all
possible permutations of the vertices, which
also permutes the triples (α,β,γ) and
(α_1,β_1,γ_1) accordingly, but K is the same for all
permutations. Except for the Case 1(a) which was known to
Chazy <cit.> and Cases 1(b) and 3(b) which were found
in <cit.> using a different method, the remaining cases are
new to the best of our knowledge. Note that
in some cases, the second column of Table <ref>
contains a parameter other than the reciprocal of a positive integer.
The corresponding Schwarz function is then not single-valued
in its domain of definition but the Chazy XII solution y(z)
given by (<ref>) still remains single-valued. Such multi-valued
Schwarz functions are related to a single-valued Schwarz function
via a rational transformation.
For example, S(23,1k, 1k;z) in Case 2(c) is related to
the single-valued function S(12,13, 1k;z) in Case 1(a)
via a degree-2 rational transformation <cit.>

    S(12,13, 1k; ϵ z) =
    [S(23,1k,1k; z)-2]^2/4[1-S(23, 1k,1k; z)] ,     ϵ = √(-14) .

In this case
the associated triangle in the z-plane with interior angles
{2π3, πk, πk} is divided along
the bisector of the angle 2π3 into two triangles each with
interior angles {π2, π3, πk}.
We shall discuss such rational transformations among the Schwarz
triangle functions in Section <ref>, which correspond to decomposition
of a curvilinear triangle into two or more similar sub-triangles. Each of the cases
marked with an asterisk corresponds to a single valued triangle function
but the parameter K is different from that considered by Chazy.
Each such case can be transformed to one in the immediately
preceding row by a rescaling k →km for m=2,3 or 4.

It is useful to note[We thank a referee for this observation.] that if
one rescales the Chazy function by
introducing Y(z) = y(z)/6 then (<ref>) takes the form

    Y”'- 12YY”+3Y' ^2 = K'(Y'-Y^2)^2,      K'=0         
    K'=864/36-k^2.

Then using (<ref>) and (<ref>), Y(z) can be expressed
as a convex linear combination of the gDH variables as follows

    Y(z) = β_1ω_1+α_1ω_2+γ_1ω_3 ,     α_1+β_1+γ_1=1 ,

where the possible values of (α_1,β_1) are listed in Table <ref>.



 §.§ Chazy XII and 3 × 3 systems

We now make a couple of observations regarding the equivalence
between the Chazy XII equation and systems of first order equations.
The first is the relationship between the Chazy XII solution y(z) and the gDH variables w_i given by (<ref>).
Note that for each case in Table <ref>,
there is a gDH system (<ref>)
defined by the triple (α, β, γ). Then from (<ref>)
with coefficients given by a_1 = 6β_1, a_2=6α_1 and
a_3=6γ_1, this gDH system can be reduced to the Chazy XII
equation (<ref>) with the corresponding Chazy parameter K listed
in Table <ref>. It should be evident from Table <ref> that
there are several gDH systems corresponding to the same Chazy XII
equation. This fact is illustrated in Table <ref>, where we list
the combinations of the gDH variables w_i which give the same y(z)
satisfying the Chazy XII equation with parameter K=436-k^2.
However, in each case the w_i satisfy the gDH system (<ref>)
with a different function τ^2 parameterized by the triple
(α,β,γ) listed in the corresponding row of Table <ref>.


The last column of Table <ref> lists the function ϕ(z) whose
logarithmic derivative in (<ref>) gives the solution y(z) for the Chazy XII equation. Since ϕ(z) are expressed in terms
triangle functions S(α,β,γ;z) and their derivatives,
it is possible for the same solution y(z) to be parametrized by
two different triangle functions s_1(z) and s_2(z). For instance,
Cases 1(a) and 1(b) correspond to s_1(z)=S(,13,1k; z)
and s_2(z)=S(13,13,2k; z). Then it follows from (<ref>)
that the correspon­ding ϕ(z) must be proportional. This leads to a differential relation between the two triangle functions, namely,

    s_1'(z)/s_1^1/2(s_1-1)^2/3 = C s_2'(z)/s_2^2/3(s_2-1)^2/3 ,

for some constant C, and can be solved to yield
s_1(z) = (2s_2(ϵ z)-1)^2, ϵ=C4^-1/3.
In fact, all the triangle functions listed in Table <ref> are related via
rational transformations, which will be deduced in the next section
by employing certain well-known transformations between solutions
of hypergeometric equations.

Lastly, we introduce a Ramanujan-type system that is equivalent
to the Chazy XII equation. Recall
from Section <ref> that the Ramanujan system (<ref>) is equivalent
to the Chazy III equation given by (<ref>) with K=0 if one
identifies y(z)=π i P(q). It will be shown that both (<ref>) and the
differential system (<ref>) introduced below have a simple geometric
interpretation. Using the notations in Section <ref>, let us define

    P̂(z) := 3/π iη(z),     Q̂(z) := 18/π^2Ω(z),
        R̂(z) := 27i/π^3∇Ω(z) .

Then, if η(z) satisfies (<ref>),
the triple (P̂, Q̂, R̂) satisfies the differential system

    1/2 π iP̂'(z) =P̂^2-Q̂/12,     1/2 π iQ̂'(z)=P̂Q̂-R̂/3,     1/2 π iR̂'(z) =P̂R̂/2-J/24Q̂^2.

The first two equations in (<ref>) are simply the definitions
of the quadratic differential Ω=η'-η^2/2 and its covariant
derivative ∇Ω = Ω' - 2ηΩ associated with the
affine connection η(z). The third one is
equation (<ref>) which is equivalent
to the Chazy XII equation with J=12-108K, K being
the Chazy parameter. When J=12, (<ref>) reduces to the Ramanujan
system (<ref>) which is equivalent to the Chazy III equation.
More explicitly, if y(z) is a solution of (<ref>)
then

    P̂= y/π i,     Q̂ = - 6y'-y^2/(π i)^2 ,
        R̂ = 9y”-9yy'+y^3/(π i)^3

satisfy (<ref>) which subsumes the Ramanujan system (<ref>).
The Ramanujan triple (P, Q, R) has a modular interpretation since
it is related to the Eisenstein series associated with the modular
group SL_2(ℤ). The functions P̂, Q̂, R̂ are also automorphic
forms for the triangle group Γ and are parameterized by the triangle
functions S(α, β, γ; z) via (<ref>). In fact
they transform as follows:

    P̂(γ(z)) = (cz+d)^2 P̂(z) + 6c(cz+d)/π i ,
          Q̂(γ(z)) = (cz+d)^4Q̂(z) , 
    R̂(γ(z)) = (cz+d)^6R̂(z) ,     where    γ(z) = az+b/cz+d,     [ a b; c d ]∈Γ .

However, we are not aware of any deep automorphic interpretation
for (P̂, Q̂, R̂)
similar to that of the Ramanujan triple.
Ramanujan <cit.> also gave
a parametrization of his triple using the hypergeometric
function _2F_1(,,1; s) that is related to the
complete elliptic integral of the first kind. In the following section,
we discuss the parametrizations of the Chazy XII solution y(z) as
well as (P̂, Q̂, R̂) in terms of
hypergeometric functions.



§ PARAMETERIZATION OF CHAZY XII SOLUTIONS

Explicit solutions of the Chazy equation were
presented in terms of the triangle functions listed in Table <ref>
of Section <ref>. Recall that the triangle functions satisfy the
nonlinear third order equation given by (<ref>)
with the potential V(s) as in (<ref>). Equation (<ref>)
is linearized via solutions of the Fuchsian equation (<ref>) associated
with V(s) using (<ref>).
Consequently, it is more convenient to express the Chazy solution y(s(z))
implicitly, that is, in terms of the variable s and a solution u(s)
of the linear equation (<ref>). Thus it is possible to treat the
solutions of the nonlinear Chazy XII equation in terms of the classical
theory of linear Fuchsian differential equations with three
regular singular points,
equivalently, via the hypergeometric equation. This is the main purpose
of the present section.



 §.§ Hypergeometric parametrization

In the following the domain D of the triangle functions s(z) will be taken
as the interior of the orthogonal circle C discussed in Section <ref>,
and the hypergeometric form of
the Fuchsian differential equation (<ref>) will be considered, in order
to make contact with standard literature.
If u(s) is a solution of (<ref>), then the function

    χ(s) = s^(α-1)/2(s-1)^(β-1)/2u(s)

satisfies the hypergeometric equation


    χ”+(1-α/s+1-β/s-1)χ'+
    (α+β-1)^2-γ^2/4s(s-1)χ=0 ,

which can be written in more standard form as

    s(s-1)χ”+[(a+b+1)s-c]χ'+abχ=0,


where a=(1-α-β-γ), b=(1-α-β+γ),
and c=1-α. The transformation (<ref>) sets the local exponents
to (0, α) at s=0, (0, β) at s=1 and
((1-α-β-γ), (1-α-β+γ)) at s=∞.
Note, however, that the exponent differences
as well as the ratio z(s) of any two linearly independent solutions
of (<ref>) coincide with those for (<ref>). Consequently,
one can employ the ratio of two independent solutions of (<ref>) instead of (<ref>) to construct the
conformal mapping and triangle function described in Section <ref>.

We next outline how to construct the triangle functions s(z)
listed in Table <ref> together with their orthogonal circles using pairs
of linearly independent hypergeometric solutions (see, e.g., <cit.>).
Notice from (<ref>) that z(s) is defined up to an arbitrary Möbius
transformation with three complex parameters of the z-plane
depending on the choice of linearly independent solutions
χ_1 and χ_2 of (<ref>).
One can then choose two of the three parameters in the Möbius transformation
in such a manner that the conformal map (<ref>) results in a triangle which has the vertex z(0) placed
at the origin of the z-plane, and the two circular arcs meeting there
can be transformed to linear segments subtending angle πα
at this vertex z(0). The remaining freedom in the Möbius transformation
can be used to rotate the line segment connecting the vertices
z(0) and z(1) onto the real axis. The remaining side
joining z(1) and z(∞) is formed by the arc of a circle such that
the origin O is in the exterior of this circle when the sum of the interior
angles of the triangle is less than π, i.e., when α+β+γ < 1.
Hence, it is possible to draw a line from the origin tangent to this circle
at some point F (see Fig. <ref>, Appendix <ref>). Then there exists
a unique circle C with center
at the origin O and passing through the point F, thus having radius OF,
that is orthogonal to the straight edges of the triangle
thus constructed. Consequently, C is orthogonal to each side of the
triangle.

A pair of hypergeometric solutions whose ratio
maps the upper half s-plane onto a triangle constructed above is
given by χ_1 =_2F_1(a,b;c;s) and χ_2 = s^1-c _2F_1(a-c+1,b-c+1;2-c;s).
Here the notation of (<ref>) has been used and _2F_1(a,b;c;s) is
the standard hypergeometric series solution of (<ref>), analytic
in the neighborhood of s=0 with _2F_1(a,b;c;s=0)=1. Note that
χ_2 vanishes at the branch point s=0 if α=1-c >0.
The explicit form of the map is then

    z(s) = χ_2/χ_1=s^1-c _2F_1(a-c+1,b-c+1;2-c;s)/_2F_1(a,b;c;s) .

Since the triples (α, β, γ)
listed in Table <ref> satisfy (<ref>), α >0.
Hence, it is clear that z(0)=0.
Furthermore, the parameters (a,b,c) are real
and positive. Consequently, the _2F_1 functions are also real and positive
for Re(s)>0. Moreover, if we take the real, positive branch
of s^1-c when Re(s)>0, then it follows from (<ref>) that z(s) is also real and positive as s varies from 0 to 1 along
the real s-axis. This confirms that one side of the triangle lies
on the positive z-axis joining the vertices z(0)=0 and z(1).
The other side of the triangle originating from
the vertex z(0)=0 is the conformal image of the negative real s-axis
on which the _2F_1 functions are also real but the factor
s^1-c=(-|s|)^1-c =|s|e^iπα. This shows that the negative
real s-axis is mapped to a linear segment joining z(0) and z(∞),
and making an angle πα with the positive z-axis.
One can also compute the vertices z(1) and z(∞) by considering
the analytic continuations of the _2F_1 functions into the neighborhoods
of s=1 and s=∞. These are given by (see, e.g., <cit.>)

    z(1)=Γ(2-c)Γ(c-a)Γ(c-b)/Γ(c)Γ(1-a)Γ(1-b),     
    z(∞) = e^π i (1-c)Γ(b)Γ(c-a)Γ(2-c)/Γ(c)Γ(b-c+1)Γ(1-a) ,

where Γ(·) is the Gamma function. From the expressions for z(1)
and z(∞), it is possible to determine the radius R of the
orthogonal circle C which forms the natural barrier beyond which s(z)
can not be analytically continued. In terms of the triple (α, β, γ)
parameterizing the triangle function S(α,β,γ;z) the square of the
radius of the barrier is <cit.>

    R^2 = (Γ(1+α)/Γ(1-α))^2
    ∏_ϵ_i=± 1Γ((1-α+ϵ_1β+ϵ_2γ))/Γ((1+α+ϵ_1β+ϵ_2γ)) .

The above expression appears in <cit.> but without a derivation,
which, although elementary, is not immediately obvious. For that reason, we
have included a brief derivation in Appendix <ref>.

The map z(s) in (<ref>) is a Puiseux series in s of the
form z(s) = s^αψ(s) where ψ(s) is analytic near s=0
with ψ(0) ≠ 0. In fact, the power series for ψ(s) can
be readily derived from the series expansion of the _2F_1 functions
in (<ref>). Since α = 1n, n ≥ 2, n ∈ℕ,
the series for z(s) can be inverted to obtain the power series
of the inverse s(z) in the form

    s(z)= z^n(1+b_1z^n+b_2z^2n+⋯) ,

where the coefficients b_j can be obtained recursively from the
coefficients in the series expansion of ψ(s). The series for s(z)
converges in a neighborhood of z(0)=0 and defines a single-valued, holomorphic
function in this neighborhood as discussed earlier in Section <ref>.
By analytic continuation of the
hypergeometric functions onto the neighborhoods of s=1 and s=∞,
it is possible to obtain similar series expansions for s(z) near
z(1) and z(∞) as well. Note that s(z) has a pole of order
1γ at z(∞).

Let (χ_1, χ_2) be the pair of hypergeometric solutions
whose ratio defines z(s) as in (<ref>), then
s'(z) = 1/z'(s)=χ_1^2/W(χ_1, χ_2)
where the Wronskian of the pair of solutions
W(χ_1, χ_2) = As^α-1(s-1)^β-1
from Abel's formula. The nonzero constant A is found by explicitly
calcu­la­ting the Wronskian of (χ_1, χ_2) in
(<ref>) and letting s → 0. Thus one obtains A = (-1)^1-βα.
Substituting the expression for s'(z) in (<ref>) yields
a parametrization for y(z) in terms of χ_1(s) and χ_1'(s), namely

    y(s(z)) = 3/α s^1-α(1-s)^1-β(2χ_1χ_1'+[α_1-α/s+β_1-β/s-1]
    χ_1^2) .

To be clear y(z) is expressed parametrically by z(s) in (<ref>) and y(s) given by (<ref>).

Note that when α=α_1, β=β_1, y(s(z)) is analytic
at the vertices z=z(0), z(1) since χ_1(s) is analytic
at s=0, 1. On the other hand, if α=α_1 but β≠β_1
then y(s(z)) is analytic at z(0) but has a simple pole at z=z(1). These
observations are consistent with Cases 1 and 2 in Section <ref>.

The Ramanujan-type triple (P̂, Q̂, R̂) of automorphic
functions introduced in Section <ref> can also be formulated in terms of
the hypergeometric function χ_1(s) =_2F_1(a,b;c;s).
The function P̂=y/(π i) is obtained directly from (<ref>) above.
In order to obtain expressions for Q̂, R̂, one first recalls
from Section <ref> that the quadratic differential Ω and its
covariant derivatives are given in terms of s(z) and s'(z) by

    Ω(s(z)) = -V_2(s(z))s'(z)^2 ,     ∇Ω(s(z)) = -V_3(s(z))s'(z)^3 ,

where the rational functions V_2(s) and V_3(s) are obtained from (<ref>).
Then from (<ref>) together with the relation s'(z) =χ_1^2/W(s) it follows that

    Q̂ = -2(3/π W(s))^2V_2(s)χ_1^4(s) ,     R̂ = (3i/π W(s))^3V_3(s)χ_1^6(s),

where W(s)=α s^α-1(1-s)^β-1.
The explicit form of the function V_2(s)
is given in (<ref>), and from (<ref>) one obtains
V_3(s)=V_2'(s)+(1-α_1s+1-β_1s-1)V_2(s). Alternatively,
one could use (<ref>) and the first two equations from (<ref>) to derive (<ref>).

Equations (<ref>) and (<ref>) provide parametrizations
of the triple (P̂, Q̂, R̂), equivalently the differential
geometric quantities (η, Ω, ∇Ω) in terms of s
and the hypergeometric function χ_1(s). Conversely,
it is also possible to derive expressions
for S(α,β,γ;z) and χ_1
in terms of the functions P̂, Q̂ and R̂.
For instance, Case 1(a) in Table <ref> gives rise to the following relations

    (Jπ^2/12Q̂)^1/4 =
    (1-s)^1/12 _2F_1(112-12k,112+12k;; s),
    
    
    s(z)= S(,13;1k; z) = 12 R̂^2/12 R̂^2-JQ̂^3.

Note that if k →∞ (Chazy III case with J=12),
one recovers the well known representation of the modular function
S(,13;0; z) from above in terms
of the Ramanujan functions Q and R, which are the Eisenstein series
of weight 4 and 6 respectively, for the modular group. Moreover, the
first relation above leads to (a slight variant of) a remarkable
identity discovered by Ramanujan <cit.>

    √(π)Q^1/4 = _2F_1(112,512;; R^2Q^3) .

This identity is obtained by first letting k →∞ and then applying
the Pfaff transformation (1-s)^1/12 _2F_1(112,112;; s)
= _2F_1(112,512;; ss-1).



 §.§ Pull-back maps of Schwarz functions

Recall that Table <ref> of Section <ref> lists the parametrizations
of the Chazy XII solution y(z) with Chazy parameter 436-k^2
in terms of different Schwarz triangle functions.
It follows from (<ref>) that any two of these Schwarz functions
are related via the differential relation
ϕ_1(z) = C ϕ_2(z) for some constant C, where the ϕ_i(z)
are listed in the last column of Table <ref>. In fact, the
example given below Table <ref> provides such a mapping between the triangle
functions corresponding to Cases 1(a) and 1(b) of Table <ref>. In this
subsection we systematically outline the transformations among the various
triangle functions linking all the cases presented in Table <ref>.

The mappings of the Schwarz functions stem from the well-known algebraic
transformations of the hypergeometric functions induced by
the pull-back transformation of the corresponding hypergeometric differential
equation (<ref>) <cit.>. These transformations are of the form
_2F_1(a,b;c ; s) = ξ(s) _2F_1(a',b'; c'; θ(s)), where
θ(s) is a rational function, and ξ(s) is an algebraic
function (see, e.g., <cit.>, also <cit.> for more recent
work). To make this paper self-contained,
we include here the details of the derivations of the relations between the
triangle functions listed in Table <ref>. It suffices to derive the
transformations relating Case 1(a) to all other cases in Table <ref>.
Let s̃ = θ(s) denote the rational map where
s̃=S(, 13, 1k;z) corresponds to Case 1(a)
and s=S(α, β, γ ;z) corresponds to any of the
other cases. Let s̃=f_1(s_1)=f_2(s_2) denote
two such rational maps for Schwarz functions s_1(z) and s_2(z)
which parametrize the same Chazy XII solution y(z). Then the
transformation between s_1 and s_2 can be expressed as
s_2 = (f_2^-1∘ f_1)(s_1), which may generally
not be single valued.

Below we consider each case from Table <ref> separately,
treating Case (i) below as an illustrative example of the procedure.

Case (i). The mapping between the Schwarz functions
S(, 13, 1k; z) and S(13, 13, 2k; z)
corresponding to Cases 1(a) and 1(b) of Table <ref>
follows from the well-known quadratic transformation <cit.>

    _2F_1(a,b;c; s) = _2F_1(a2,b2; c; 4s(1-s)),
         c = a+b+1/2,

of hypergeometric functions. Note that the parameters (a,b,c) above
are constrained by a linear relation.
Choosing a=16 - 1k, b =16 + 1k
whence c=23, yields (α, β, γ)=(1-c, c-a-b, b-a)
= (13, 13, 2k) and (α, β, γ) =
(1-c, c-(a+b), (b-a)) = (13, 12, 1k). These values
correspond to the exponent differences at the singular points
s=0,1,∞ (up to their permutations).
One applies the quadratic transformation to the
_2F_1 functions in the right hand side of the map (<ref>)
to construct z(s) first, and then invert the obtained relation
to recover the map between the two triangle functions. Thus, in this case,
one obtains from (<ref>)

    z(s) = s^1/3_2F_1(12-1k,12+1k; 43; s)/_2F_1(16 - 1k,16 + 1k; 23; s) = (s(1-s))^1/3_2F_1(56 - 1k,56 + 1k; 43; s)/_2F_1(16 - 1k,16 + 1k; 23; s) ,

where the Euler transformation
_2F_1(a,b;c; s)=(1-s)^c-a-b _2F_1(c-a,c-b;c; s) and the symmetry
_2F_1(a,b;c; s)=_2F_1(b,a;c; s) have been used to obtain the
last equality. Note that the parameters in the _2F_1 functions
of this last expression satisfy the relation c=(a+b+1)/2.
Next, the quadratic transformation is applied
to both the _2F_1 functions on the numerator and denominator to yield

    z̃(s̃):=√(4)z(s) = s̃^1/3_2F_1(512 - 12k,512 + 12k; 43; s̃)/_2F_1(112 - 12k,112 + 12k; 23; s̃)    s̃:=4s(1-s) .

Inverting the above relation leads to s̃ = 4s(1-s) where
s̃(z) = S(13, 12, 1k; √(4)z) and
s(z)= S(13, 13, 2k; z). By making the substitution
s̃→ 1-s̃, which amounts to switching the singular
points s̃=0 and s̃=1, one obtains

    s̃=θ(s)=1-4s(1-s)=(2s-1)^2,     s̃(z)= S(12, 13, 1k; √(4)z),

as the rational map θ(s) of degree 2 between the two triangle
functions listed in Cases 1(a) and 1(b) of Table <ref>.
The associated pull-back transformation is denoted in terms of
the local exponents as
(12, 13, 1k) 2 (13, 13, 2k) <cit.>.

Case (ii). Here, the transformation between the Schwarz function
for each of the three subcases in Case 2 of Table <ref>
and the Schwarz function in Case 1(a) will
be discussed. The mapping between Cases 2(a) and 1(a) is obtained
from the cubic transformation of hypergeometric equation <cit.>

    _2F_1(a,1-a3;4a+56;s) =
    (1-4s)^-a _2F_1(a3,a+13;4a+56; s̃),     s̃=27s/(4s-1)^3,

where the _2F_1 parameters (a,b,c) depend only on a. Choosing
a=14-32k, the above transformation gives the degree 3 map
(1k, 12, 13) 3 (1k, 12, 2k).
In this case, (<ref>) leads to

    z(s) = s^1/k_2F_1(14+32k,14-12k;1+1k;s)/_2F_1(14-32k,14+12k;1-1k;s) ,

which, after applying the cubic transformation of the hypergeometric
functions, leads to

    z̃(s̃) := (-27)^1/kz(s)=s̃^1/k_2F_1(112+12k,512+12k;1+1k;s̃)/_2F_1(112-12k,512-12k;1-1k;s̃).

Inverting the above relation, one obtains the transformation
between the triangle functions
s̃(z)=S(1k,12,13;z̃)=27s/(4s-1)^3, s(z)=S(1k,12,2k;z). In order to obtain the degree 3 map
relating Case 2(a) to 1(a), one needs
to switch the singular points s=0 and s=1 so that s → 1-s
in above, and also permute the singular points for s̃
as (0,1,∞) → (∞, 0, 1) by letting s̃→ 1-s̃^-1.
Thus, one obtains the degree 3 map
(12, 13, 1k) 3 (12, 1k, 2k) given by

    s̃=S(12, 13, 1k; z̃) = s(8s-9)^2/27(1-s),     
    s(z) = S(12, 1k, 2k;z) .


The map between the Schwarz functions for Cases 2(b) and 1(a) of Table <ref> is
obtained via a quartic transformation of the _2F_1 functions <cit.>

    _2F_1(a,a+13;3a+56; s) = (1+8s)^-3a/4 _2F_1(a4,a4+13;3a+56; s),     s̃=64s(1-s)^3/(1+8s)^3 .

Choosing a=13-2k, the above transformation induces a degree 4 map
(1k, 12, 13) 4  (1k, 3k, 13). Then
(<ref>) after applying the Euler transformation to the
_2F_1 function in the denominator, yields the following

    z(s) = [s(1-s)^3]^1/k_2F_1(13+2k,23+2k;1+1k;s)/_2F_1(13-2k, 23-2k;1-1k; s) .

Applying the quartic transformation to the _2F_1 functions in the
above expression, one obtains

    z̃(s̃) = (64)^1/kz(s) = s̃^1/k_2F_1(112+12k,512+12k;1+1k;s̃)/_2F_1(112-12k, 512-12k;1-1k; s̃) .

Then the substitutions s → (1-s)^-1 and s̃→ 1-s̃^-1
yield the degree 4 map
(12, 13, 1k) 4 (13, 1k, 3k)
between Cases 2(b) and 1(a), given by

    s̃(z)=S(12, 13, 1k; z̃) = (27s^2-36s+8)^2/64(1-s),
         s(z) = S(13, 1k, 3k;z) .


The last subcase involving the mapping between Cases 2(c) and 1(a)
of Table <ref> is derived once again from a quadratic transformation of
hypergeometric functions. It is a degree 2 map
(12,13,1k) 2 (23,1k,1k) given by

    s̃=S(12,13,1k;z̃) = (s-2)^2/4(1-s),     
    s(z) = S(23,1k,1k;z) ,     z̃ =(-14)^1/3 z .

It is obtained by employing the quadratic transformation
discussed in Case (i) with hypergeometric parameters
(a,b,c)=(16-1k,56-1k,1-1k), and then making the substitutions
s → s^-1, s̃→ 1-s̃^-1. Alternatively, one can
start from the quadratic transformation <cit.>

    _2F_1(a,b;2b;s) = (1-s)^-a/2_2F_1(a2,b-a2;b+12; s̃) ,     s̃=s^2/4(s-1) ,

with parameters (a,b,c)=(16-1k,16,13), then apply the
map s̃→ 1-s̃.

Case (iii). There are two subcases for Case 3 in Table <ref>.
For both of these subcases the pull-back
map to Case 1(a) of Table <ref> is of degree 6, which can be expressed
as a composition of a degree 2 and a degree 3 map. Specifically,
the map between cases 1(a) and 3(a) can be schematically represented as

    (12,13,1k) 3 (12,1k,2k)
    2 (1k,1k,4k) .

The degree 2 map above is given by θ_1(s)=(2s-1)^2 as in Case (i) above.
The degree 3 map given by θ_2(s)= s(8s-9)^2/27(1-s), is the same as
the one between Cases 1(a) and 2(a) of Table <ref> and
derived in Case (ii). Thus, for this
case, s̃=θ_2 ∘θ_1(s).

Similarly, Case 3(b) is mapped to Case 1(a) of Table <ref> according to

    (12,13,1k) 2 (13,13,2k)
    3 (2k,2k,2k) .

The degree 3 map transforming Case 3(b) to the intermediate step
(Case 1(b) of Table <ref>) follows from a different cubic transformation
from the one in Case (ii). It is given by <cit.>

    _2F_1(a,a+13; 2a+23;s)=
    (1+ω^2s)^-a _2F_1(a3, a+13;2a+23;s̃),
        s̃=As(s-1)/(s+ω)^3 ,

where A=3(2ω+1) and ω=e^2π i/3.
Choosing a=-3k and proceeding as in the previous cases,
yields the map (2k,13,13) 3 (2k,2k,2k). Then after the substitution
s̃→s̃^-1, one obtains

    s̃(z)=S(13,13,2k; (-A)^2/kz) =
    (s+ω)^3/As(s-1) := θ_1(s),     
    s(z)=S(2k,2k,2k;z) .

The degree 2 map θ_2(s) (12,13,1k) 2
 (13,13,2k) was already discussed in Case (i). Hence, the
composition map s̃=θ_2 ∘θ_1(s) gives
the transformation between the Schwarz functions
in Case 3(b) and Case 1(a) of Table <ref>.

It can be readily verified that in all the above cases, the rational
map s̃=θ(s) is a solution of the differential relation
ϕ(s̃) = ϵϕ(s) for constant ϵ,
for the functions ϕ listed the last column of Table <ref>.
An example involving the Cases 1(a) and 1(b) was presented earlier
in Section <ref> (below Table <ref>). Explicitly, this differential
relation reads as

    dx =  ds̃/s̃^1/2(s̃-1)^2/3 =
     ϵ ds/s^1-α_1(s-1)^1-β_1 ,

where s̃(z) corresponds to the Schwarz function in Case 1(a)
and s(z) represents the Schwarz function for each of the other
cases in Table <ref>. It follows from our discussion
in Section <ref> that, geometrically, the above differentials
define the “flat” coordinate x on a one-dimensional manifold
M ⊆ℂℙ^1 in which the affine connection
η associated with the Chazy XII solution y(z) is trivialized.
Both s and s̃=θ(s) are projective invariant coordinates
on M. It is interesting to note that solving the above
differential relation directly provides an alternative way to derive
the map θ(s), instead of using the
algebraic transformations of the hypergeometric functions. However,
we do not pursue this direct approach here.
We also remark that the differential ϕ(z)dz is a primitive of the incomplete
beta integral (see, e.g., <cit.>)

    B(α_1, β_1; s) = ∫_0^st^α_1-1(t-1)^β_1-1  dt
    = s^α_1/α_1 _2F_1(α_1, 1-β_1;α_1+1;s) ,

which is also related to the hypergeometric functions. Thus, the rational
map s̃= θ(s) corresponds to algebraic transformation of the beta
functions. The derivation of these rational maps directly from
the solutions of (<ref>) have not been studied to the best
of our knowledge.



§ CONCLUDING REMARKS

In this article, we have presented a method that converts the
Chazy XII differential equation into an algebraic relation.
This method is based on a simple differential geometric interpretation
of the Chazy XII equation given by Dubrovin in <cit.>.
By the way of elucidating this algebraic relationship, we have derived all possible
parametrizations of the Chazy XII solution y(z) via the Schwarz triangle
functions S(α, β, γ;z) for α+β+γ<1.
Our method can be extended in a straightforward way to the
case when α+β+γ≥ 1 but we do not pursue it here.
Furthermore, we show that these parametrizations can be described
in terms of classical hypergeometric functions. Using the hypergeometric
theory, the center and radius of the natural barrier for the Chazy XII solution
are found explicitly. Finally, the algebraic transformations of
the hypergeometric functions are used to obtain rational maps between
the Schwarz functions corresponding to the Chazy XII solution.

It is known that the Chazy III equation is equivalent to the Ramanujan
differential relations (<ref>) for the Eisenstein series P, Q, R
for the modular group SL_2(ℤ). Ramanujan also derived a number of remarkable
identities among the functions P, Q, R using their representation in terms
of hypergeometric functions. Likewise, we have introduced a Ramanujan-like
triple (P̂, Q̂, R̂) which satisfy differential relations
that are equivalent to the Chazy XII equation. Additionally, these
functions also satisfy interesting identities that can be derived from their
hypergeometric representations. We have presented an example at the
end of Section <ref>, but a comprehensive exploration of this line of
investigation is for future study.

The rational maps between the Schwarz functions using the pull-backs
of hypergeometric equations were presented in Section <ref>. However, we
found that these maps can be also obtained by solving certain first
order differential relations among the Schwarz functions. These relations
follow directly from the representation of the Chazy XII solution y(z) in terms of the function ϕ(z) in (<ref>). We believe
that all rational maps of Schwarz functions satisfy such differential relations
although we have not pursued this question further here since
it is beyond the scope of this article. We plan to study this
issue in a future work.




§ PROOF OF LEMMA <REF>

In this appendix we prove Lemma <ref> from Section <ref>.

Suppose J=0. Then (<ref>) and (<ref>) lead to 4 cases:

    (1) A = B=0,      (2) α_1=β_1=0,      (3)  A=β_1=0,
         (4) B=α_1=0 .

When A=B=0, the remaining equations yield the following conditions
on α_1, β_1 and C

    (1-2α_1)(1-3α_1)C=(1-2β_1)(1-3β_1)C
     =
    [(1-2α_1)(2-3β_1)+ (1-2β_1)(2-3α_1)]C=0 .

If C=0, then A=B=C=0 implies α=α_1, β=β_1,
and γ=γ_1 from the definitions of A, B, C given
in Section <ref>.
Therefore, α+β+γ=α_1+β_1+γ_1=1, which violates
the condition α+β+γ<1 in (<ref>). Hence, C ≠ 0.
Then the first two conditions above imply

    α_1 ∈{, 13},     β_1 ∈{, 13}.

Except for the case
{α_1, β_1} = {, }, the remaining values of
α_1 and β_1 do not satisfy the third condition given above.
On the other hand, if {α_1, β_1} = {, }, then
α=α_1= and β = β_1= because A=B=0.
Hence, α+β = 1, which contradicts (<ref>).

If α_1=β_1=0, then γ_1=1. In this case, (<ref>) gives
A+B+C=0, which implies γ = γ_1 =1, again violating (<ref>).

Cases (3) and (4) are interchangeable by switching A with B, and
α_1 with β_1. So it suffices consider only one case, say if
A=β_1=0. Then (<ref>) implies

    (1-α_1)B +  C=0

Eliminating B from (<ref>) by using (<ref>) leads to
the condition (1-2α_1)C=0. If α_1=, (<ref>)
implies that B+C=0 which is equivalent to γ=γ_1=
after taking into account A=0 and β_1=0. Hence, α=γ=,
which contradicts α+β+γ<1.

Finally, if C=0 in (<ref>), then either α_1=1, or B=0.
The case α_1=1 is impossible because then A=0 would imply
α=α_1=1, which does not satisfy (<ref>).
If B=0, then A=B=C=0. This is impossible as argued
in Case (1) above.




§ RADIUS OF THE ORTHOGONAL CIRCLE

In this appendix we briefly outline a derivation of
the expression in Section <ref> for the radius R
of the orthogonal circle C, which forms the natural barrier in the
complex plane to the solutions of (<ref>).

Recall from Section <ref> that (<ref>) maps the upper-half s-plane
onto a circular triangle with two straight edges denoted by OA and OB as shown in Fig. <ref>.
The third side of the triangle is formed by the circular arc AB which
is part of the circle Q with center at D, and radius AD= BD:=r.
The vertices O, A, B correspond to the image points z(0)=0,
z(1) and z(∞) of the map (<ref>), respectively.
The angle πα=∠ BOA, while angles πβ and πγ are
the angles made by OA and OB respectively, with the circular arc AB.
The orthogonal circle C is centered at the origin O and its radius
OF:=R. Now consider the triangles OAB and DAB whose common side is
given by the chord AB. From elementary considerations, it follows that
∠ ADB=π-π(α+β+γ), ∠
DAB=π(α+β+γ2), and
∠ OBA=π2-π(α+β-γ2).
Applying the law of sines to triangles OAB and DAB, and denoting
OA=x, one finds that

    AB = x sinπα/cosπ(α+β-γ2),
        BD:= r = ABsinπ(α+β+γ2)/sinπ(α+β+γ) .

Eliminating AB from above, yields the following expression for the radius
of the circle Q

    r=x sinπα/2cosπ(α+β-γ2)
    cosπ(α+β+γ2) .

Since the circles Q and C are mutually orthogonal, their radii
OF and DF are perpendicular. Hence from the right triangle ODF
with hypotenuse OD=:d it follows that R^2=d^2-r^2. Next, considering the
triangle OAD with ∠ OAD=π2+πβ, one obtains
d^2=x^2+r^2+2rxsinπβ. Substituting this into the expression
for R^2 and using (<ref>), one obtains

    R^2 = x^2 sinπ2(1-α+β-γ)
    sinπ2(1-α+β+γ)/sinπ2(1-α-β+γ)
    sinπ2(1-α-β-γ) ,

after using some trigonometric identities. From (<ref>) the line segment
OA=x along the real axis is given by

    x=z(1)=_2F_1(a-c+1, b-c+1; 2-c; 1)/_2F_1(a, b; c; 1)
    =Γ(2-c)Γ(c-a)Γ(c-b)/Γ(c)Γ(1-a)Γ(1-b) ,

after using _2F_1(a, b; c; 1)=Γ(c)Γ(c-a-b)[Γ(c-a)Γ(c-b)]^-1 <cit.>.
Substituting the above expression for x into (<ref>),
using the identity Γ(t)Γ(1-t)=π csc(π t), and replacing
(a, b, c) by their values in terms of (α, β, γ), one finally
obtains the expression for R^2 in Section <ref>.



 §.§ Acknowledgments

The work of SC was partly supported by NSF grant No. DMS-1410862. The work of OB was supported in part by a CRCW grant from University of Colorado, Colorado Springs. The authors thank Professor Mark Ablowitz for useful discussions, as well as the anonymous referees for their valuable remarks which substantially improved the article.

[1]Referencesref
99
=0pt

Ablowitz-H1
Ablowitz M.J., Chakravarty S., Halburd R., The generalized Chazy equation and
 Schwarzian triangle functions, https://doi.org/10.4310/AJM.1998.v2.n4.a1Asian J. Math. 2 (1998),
 619–624.

Ablowitz-H2
Ablowitz M.J., Chakravarty S., Halburd R., The generalized Chazy equation
 from the self-duality equations, https://doi.org/10.1111/1467-9590.00121Stud. Appl. Math. 103
 (1999), 75–88.

Atiyah-H
Atiyah M., Hitchin N., The geometry and dynamics of magnetic monopoles,
 https://doi.org/10.1515/9781400859306M.B. Porter Lectures, Princeton University Press, Princeton, NJ, 1988.

Bureau1
Bureau F.J., Integration of some nonlinear systems of ordinary differential
 equations, https://doi.org/10.1007/BF02413618Ann. Mat. Pura Appl. 94 (1972), 345–359.

SC13
Chakravarty S., Differential equations for triangle groups, in Algebraic and
 Geometric Aspects of Integrable Systems and Random Matrices, https://doi.org/10.1090/conm/593/11866Contemp.
 Math., Vol. 593, Amer. Math. Soc., Providence, RI, 2013, 179–204.

CA96
Chakravarty S., Ablowitz M.J., Integrability, monodromy evolving deformations,
 and self-dual Bianchi IX systems, https://doi.org/10.1103/PhysRevLett.76.857Phys. Rev. Lett. 76
 (1996), 857–860.

CA09
Chakravarty S., Ablowitz M.J., Parameterizations of the Chazy equation,
 https://doi.org/10.1111/j.1467-9590.2009.00463.xStud. Appl. Math. 124 (2010), 105–135, https://arxiv.org/abs/0902.3468arXiv:0902.3468.

CAC90
Chakravarty S., Ablowitz M.J., Clarkson P.A., Reductions of self-dual
 Yang–Mills fields and classical systems, https://doi.org/10.1103/PhysRevLett.65.1085Phys. Rev. Lett.
 65 (1990), 1085–1087.

CAT92
Chakravarty S., Ablowitz M.J., Takhtajan L.A., Self-dual Yang–Mills
 equation and new special functions in integrable systems, in Nonlinear
 Evolution Equations and Dynamical Systems (Baia Verde, 1991), World Sci.
 Publ., River Edge, NJ, 1992, 3–11.

Chazy3
Chazy J., Sur les équations différentielles du troisième ordre et d'ordre
 supérieur dont l'intégrale générale a ses points critiques fixes,
 https://doi.org/10.1007/BF02393131Acta Math. 34 (1911), 317–385.

CO1996
Clarkson P.A., Olver P.J., Symmetry and the Chazy equation,
 https://doi.org/10.1006/jdeq.1996.0008J. Differential Equations 124 (1996), 225–246.

Cosgrove
Cosgrove C.M., Chazy classes IX–XI of third-order differential equations,
 https://doi.org/10.1111/1467-9590.00134Stud. Appl. Math. 104 (2000), 171–228.

Darboux
Darboux G., Mémoire sur la théorie des coordonnées curvilignes, et des
 systèmes orthogonaux, https://doi.org/10.24033/asens.159Ann. Sci. École Norm. Sup. (2) 7
 (1878), 101–150.

Dubrovin
Dubrovin B., Geometry of 2D topological field theories, in Integrable
 Systems and Quantum Groups (Montecatini Terme, 1993), https://doi.org/10.1007/BFb0094793Lecture
 Notes in Math., Vol. 1620, Springer, Berlin, 1996, 120–348,
 https://arxiv.org/abs/hep-th/9407018hep-th/9407018.

Bateman
Erdélyi A., Magnus W., Oberhettinger F., Tricomi F.G., Higher
 transcendental functions, Vol. 1, McGraw-Hill, 1953.

Ferapontov
Ferapontov E.V., Galvão C.A.P., Mokhov O.I., Nutku Y., Bi-Hamiltonian
 structure of equations of associativity in 2-d topological field theory,
 https://doi.org/10.1007/s002200050123Comm. Math. Phys. 186 (1997), 649–669.

Ford
Ford L.R., Automorphic functions, 2nd ed., Chelsea Publishing Co., New York,
 1951.

Goursat
Goursat E., Sur l'équation différentielle linéaire, qui admet pour
 intégrale la série hypergéométrique, https://doi.org/10.24033/asens.207Ann. Sci. École Norm.
 Sup. (2) 10 (1881), 3–142.

Halphen
Halphen G., Sur une système d'équations différentielles,,
 C.R. Acad. Sci. Paris 92 (1881), 1101–1103.

Halphen1
Halphen G., Sur certains système d'équations différentielles,
 C.R. Acad. Sci. Paris 92 (1881), 1404–1407.

Hitchin
Hitchin N.J., Twistor spaces, Einstein metrics and isomonodromic
 deformations, https://doi.org/10.4310/jdg/1214457032J. Differential Geom. 42 (1995), 30–112.

Hitchin1
Hitchin N.J., Hypercomplex manifolds and the space of framings, in The Geometric
 Universe (Oxford, 1996), Oxford Univ. Press, Oxford, 1998, 9–30.

Maier11
Maier R.S., Nonlinear differential equations satisfied by certain classical
 modular forms, https://doi.org/10.1007/s00229-010-0378-9Manuscripta Math. 134 (2011), 1–42,
 https://arxiv.org/abs/0807.1081arXiv:0807.1081.

Nehari
Nehari Z., Conformal mapping, McGraw-Hill Book Co., Inc., New York, Toronto,
 London, 1952.

Ramanujan-arith
Ramanujan S., On certain arithmetical functions, Trans. Cambridge
 Philos. Soc. 22 (1916), 159–186.

Ramanujan-notebook
Ramanujan S., Notebooks, Vols. 1, 2, Tata Institute of Fundamental Research,
 Bombay, 1957.

Ramanujan-collect
Ramanujan S., Collected Papers, Amer. Math. Soc., Providence, RI, 2000.

Randall16
Randall M., Flat (2,3,5)-distributions and Chazy's equations,
 https://doi.org/10.3842/SIGMA.2016.029SIGMA 12 (2016), 029, 28 pages, https://arxiv.org/abs/1506.02473arXiv:1506.02473.

Randall17
Randall M., Schwarz triangle functions and duality for certain parameters of
 the generalised Chazy equation, https://arxiv.org/abs/1607.04961arXiv:1607.04961.

Rosenhead
Rosenhead L. (Editor), Laminar boundary layers, Clarendon Press, Oxford, 1963.

Schwarz1
Schwarz H.A., Ueber diejenigen Fälle, in welchen die Gaussische
 hypergeometrische Reihe eine algebraische Function ihres vierten
 Elementes darstellt, https://doi.org/10.1515/crll.1873.75.292J. Reine Angew. Math. 75 (1873),
 292–335.

Vid09
Vidūnas R., Algebraic transformations of Gauss hypergeometric functions,
 https://doi.org/10.1619/fesi.52.139Funkcial. Ekvac. 52 (2009), 139–180,
 https://arxiv.org/abs/math.CA/0408269math.CA/0408269.



