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The general integrability cases in the rigid-body dynamics are the solutionsof Lagrange, Euler, Kovalevskaya, and Goryachev-Chaplygin. The first two can beincluded in Smale's scheme for studying the phase topology of natural systemswith symmetries. We modify Smale's program to suit the most complicated lasttwo cases with non-linear first integrals. The bifurcation sets are found andall transformations of the integral tori are described and classified. Newnon-trivial bifurcation of a torus is established in the Kovalevskaya andGoraychev-Chaplygin cases.

Topological analysis of classical integrable systems in the dynamics of the rigid body