Flat F-manifolds, Miura invariants, and integrable systems of conservation laws

We extend some of the results proved for scalar equations in Arsie et al. (2015, Nonlinearity, 28) and Arsie et al. (2015, Proc. R. Soc. A, 471, doi:10.1098/rspa.2014.0124), 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 22 found in Arsie & Lorenzoni (2017, Lett. Math. Phys., doi:10.1007/s11005-017-0963-x), we study the corresponding integrable deformations up to order 22 in the deformation parameter ϵϵ⁠. 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 apart from two values of the parameter all deformations of order one in ϵϵ are Miura-trivial, while all those of order two in ϵϵ 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