Geodesic flows on two-dimensional manifolds with an additional first integral that is polynomial in the velocities

In the paper an explicit description is given for all Riemannian metrics on the sphere and on the torus whose geodesic flows have an additional first integral that is both quadratic in the velocities and independent of the energy integral. Moreover, it is proved that on compact two-dimensional manifolds of higher genus the geodesic flows have no additional polynomial integral. All the results admit straightforward generalizations to arbitrary natural systems given on cotangent bundles of two-dimensional manifolds.
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