Integrable geodesic flows on the sphere with additional integrals of third and fourth degree in the momenta

Abstract:
The problem of searching the integrable metrics with additional integrals of higher degree on the compact two-dimensional manifolds is considered. We present the solutions of the system of overdefined differential equations for the coefficients of the first integral of third and fourth degree. Basing on some properties of the solutions, we demonstrate connection between some integrable geodesic flows and integrable many-body systems on the line, particularly the Toda lattice.