Geometrical approach to the separation of variables in mechanical systems

The article presents the analytical results received with the help of computer aided symbolic calculations in the problem of motion of the rigid body in two constant fields. The Liouville integrability of this system under certain condition of the Kowalevski type was established by A.G.Reyman and M.A. Semenov-Tian-Shansky. We consider the geometrical basis for obtaining a separation of variables. Two systems of local planar coordinates are introduced in which the projections of integral manifolds of the critical subsystems become rectangular. The separated equations are obtained for two subsystems with the help of Mathematica 7.