Аннотация:The problem of geodesic curves on a closed two-dimensional surface and some of its generalizations related with the addition of gyroscopic forces are considered. The authors study one-parameter groups of symmetries in the four-dimensional phase space that are generated by vector fields commuting with the original Hamiltonian vector field. Fields of symmetries of the first and second degree are studied. The presence of such symmetries is related to the existence of ignorable cyclic coordinates and separated variables. The influence of gyroscopic forces on the existence of fields of symmetries with polynomial components is studied.