Title: Flat $F$-manifolds, Miura invariants and integrable systems of conservation laws,
Abstract: We extend some of the results proved for scalar equations in [3,4], to the
case of systems of integrable conservation laws. In particular, for such
systems we prove that the eigenvalues of a matrix obtained from the quasilinear
part of the system are invariants under Miura transformations and we show how
these invariants are related to dispersion relations. Furthermore, focusing on
one-parameter families of dispersionless systems of integrable conservation
laws associated to the Coxeter groups of rank $2$ found in [1], we study the
corresponding integrable deformations up to order $2$ in the deformation
parameter $\epsilon$. Each family contains both bi-Hamiltonian and
non-Hamiltonian systems of conservation laws and therefore we use it to probe
to which extent the properties of the dispersionless limit impact the nature
and the existence of integrable deformations. It turns out that a part two
values of the parameter all deformations of order one in $\epsilon$ are
Miura-trivial, while all those of order two in $\epsilon$ are essentially
parameterized by two arbitrary functions of single variables (the Riemann
invariants) both in the bi-Hamiltonian and in the non-Hamiltonian case. In the
two remaining cases, due to the existence of non-trivial first order
deformations, there is an additional functional parameter.