Symmetries and the topology of dynamical systems with two degrees of freedom

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. If the genus of the surface is greater than one, then there are no nontrivial symmetries. For a surface of genus one (a two-dimensional torus) it is established that if there is an additional integral polynomial in the velocities, even or odd with respect to each component of the velocity, then there is a polynomial integral of degree one or two. For a surface of genus zero examples of nontrivial integrals of degree three and four are given. Fields of symmetries of 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.