The structure of infinitesimal symmetries of geodesic flows on a two-dimensional torus

The problem of geodesic lines on a two-dimensional torus is considered. One-parameter symmetry groups in the four-dimensional phase space that are generated by vector fields commuting with the initial Hamiltonian vector field are studied. As proved by Kozlov and Bolotin, a geodesic flow on a two-dimensional torus admitting a non-trivial infinitesimal symmetry of degree $n$ has a many-valued integral that is a polynomial of degree at most $n$ in the momentum variables. Kozlov and the present author proved earlier that first- and second-order infinitesimal symmetries are related to hidden cyclic coordinates and separated variables. In the present paper the structure of polynomial infinitesimal symmetries of degree at most four is described under the assumption that these symmetry fields are non-Hamiltonian.