On the second degree fields of symmetry with respect to impulses of geodesic flows on a two-dimensional sphere (Q1275980)

Let \(M\) be a two-dimensional Riemann manifold with Riemannian metric \(ds^2\). On \(T^*M\) a Hamiltonian system is considered which models the interval motion of a point on \(M\) with metric \(ds^2\). The vector field \(u\) on \(T^*M\) is called the field of symmetry of the Hamiltonian system \(v\), if the condition \([u,v]=0\) is satisfied, where \([\cdot,\cdot]\) is the commutator of vector fields. The main result of the paper is the following assertion. If the Gaussian curvature of the metric on the sphere is not constant, then any field of symmetry of the first and second degree is Hamiltonian.