GEODESIC FLOWS ON TWO-DIMENSIONAL MANIFOLDS WITH AN ADDITIONAL FIRST INTEGRAL THAT IS POLYNOMIAL IN THE VELOCITIES

摘要: In the paper an explicit description is given for all Riemannian metrics on sphere and torus whose geodesic flows have additional first integral that both quadratic in velocities independent of energy integral. Moreover, it proved compact two-dimensional manifolds higher genus no polynomial All results admit straightforward generalizations to arbitrary natural systems cotangent bundles manifolds. Bibliography: 8 titles.