Superintegrable Bertrand Magnetic Geodesic Flows

The problem of description superintegrable systems (i.e., with closed trajectories in a certain domain) the class rotationally symmetric natural mechanical goes back to Bertrand and Darboux. We describe all (in domain slow motions) magnetic geodesic flows. show that sufficiently motions central field on two-dimensional manifold revolution are periodic if only metric has constant scalar curvature is homogeneous, i.e. proportional area form.