Topological classification of integrable Hamiltonian systems in a potential field on surfaces of revolution

A topological classification, up to Liouville (leafwise) equivalence of integrable Hamiltonian systems given by flows with a smooth potential on two-dimensional surfaces of revolution is presented. It is shown that the restrictions of such systems to three-dimensional isoenergy surfaces can be modelled by the geodesic flows (without potential) of certain surfaces of revolution. It is also shown that in many important cases the systems under consideration are equivalent to other well-known mechanical systems.