A Family of Oscillations Connecting Stable and Unstable Permanent Rotations of a Heavy Solid with a Fixed Point

The rotation of a heavy rigid body around a fixed point is investigated if the center of gravity belongs to the main plane of the ellipsoid of inertia. A reduction of the system of Euler–Poisson equations to a reversible conservative system with two degrees of freedom is given, and a bifurcation analysis of permanent rotations is carried out. Global families of periodic motions are found that connect stable and unstable permanent rotations of the same frequency. It is proved that non-degenerate symmetric periodic motion in a reversible mechanical system always continues to the global family.