On isomorphisms of some integrable systems on a plane and a sphere

We consider trajectory isomorphisms between various integrable systems on an $n$-dimensional sphere $S^n$ and a Euclidean space $R^n$. Some of the systems are classical integrable problems of Celestial Mechanics in plane and curved spaces. All the systems under consideration have an additional first integral quadratic in momentum and can be integrated analytically by using the separation of variables. We show that some integrable problems in constant curvature spaces are not essentially new from the viewpoint of the theory of integration, and they can be analyzed using known results of classical Celestial Mechanics.