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The problem of realizability as a surface of revolution embedded into ℝ3 is studied and solved for a two-dimensional Bertrand’s Riemannian manifold being a configuration space of an inverse problem of dynamics. The problem of local realizability (near a parallel) of those manifolds is also solved.

The problem of explicit form of the metric of revolution on Bertrand’s Riemannian manifolds in particular coordinates is solved. Connections with earlier results of M. Santoprete are discussed.

Three well-known classes of surfaces of revolution are considered. The problem of their intersection and existence of common parts is studied.

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., Comment: 37 pages, 5 figures, in Russian