On some properties of semi-Hamiltonian systems arising in the problem of integrable geodesic flows on the two-dimensional torus

Bialy and Mironov demonstrated in a recent series of works that the search for polynomial first integrals of a geodesic flow on the 2-torus reduces to the search for solutions to a system of quasilinear equations which is semi-Hamiltonian. We study the various properties of this system.