Tensor invariants of natural mechanical systems on compact surfaces and the corresponding integrals (Q1569298)

A tensor field \(T\) on the unit cotangent bundle to a Riemannian manifold \(M\) is called a tensor invariant of the geodesic flow on \(M\) if the Lie derivative of \(T\) along the flow vanishes everywhere. For a scalar \(T\) this means that \(T\) is a first integral of the flow. The author classifies tensor invariants of geodesic flows on two-manifolds and proves that existence of nontrivial tensor invariants vanishing at some point implies existence of two-valued symmetry fields. He also proves the following nice theorem: if there is a nonsymplectic trajectory automorphism of the geodesic flow, then there is a nontrivial first integral.