ИСТИНА

Bertrand proved (1873) that, in Newtonian mechanics, the Kepler potential a/r + b and the harmonic oscillator potential ar^2 + b on the Euclidean plane (a > 0) are distinguished by the property that: (i) all the bounded trajectories are closed and (ii) there exist non-circular closed orbits. Natural mechanical systems possessing the above property will be called Bertrand systems. Darboux (1877) and Perlick (1992) extended the result of Bertrand by obtaining a complete description of all spherically symmetric Bertrand systems, whose underlying Riemannian manifolds of revolution have no equators. We describe all spherically-symmetric Bertrand systems, including those with equators. We also describe all rotationally-symmetric superintegrable (in a domain of slow motions) magnetic geodesic flows.