On first integrals of geodesic flows on a two-torus

For a geodesic (or magnetic geodesic) flow, the problem of the existence of an additional (independent of the energy) first integral that is polynomial in momenta is studied. The relation of this problem to the existence of nontrivial solutions of stationary dispersionless limits of two-dimensional soliton equations is demonstrated. The nonexistence of an additional quadratic first integral is established for certain classes of magnetic geodesic flows.