Integrable geodesic flows on 2-torus and the systems of hydrodynamical type

We prove that the question of existence of polynomial first integrals of the geodesic flow on 2-torus leads to a semi-Hamiltonian quasi-linear equations, i.e. the system can be written in the conservation lows form and in the hyperbolic region it has Riemannian invariants. We also prove that in the elliptic region cubic and quartic integrals are reduced to the integrals of degree one or two. The results obtained with M. Bialy.