Generalization of Bertrand's theorem to surfaces of revolution

The generalization of Bertrand's theorem to abstract surfaces of revolution without "equators" is proved. We prove a criterion for the existence on such a surface of exactly two central potentials (up to an additive and a multiplicative constants) all of whose bounded nonsingular orbits are closed and which admit a bounded nonsingular noncircular orbit. A criterion for the existence of a unique such potential is proved. The geometry and classification of the corresponding surfaces are described, with indicating either the pair of gravitational and oscillator) potentials or the unique (oscillator) potential. The absence of the required potentials on any surface which does not meet the above criteria is proved.