Iso-energetic manifolds and motion possibility regions of rigid body in double force field

The motion of a rigid body about a fixed point in a double constant force field is governed by a Hamiltonian system with three degrees of freedom. We consider the general case when there are no one-dimensional symmetry groups. We point out the critical points of the Hamilton function and corresponding critical values of energy. Using the Morse theory, we have found the smooth type of non-degenerate five-dimensional iso-energetic levels and find their projections onto the configuration space, diffeomorphic to a three-dimensional projective space. The analogs of classical motion possibility regions, the projections of iso-energetic manifolds onto one of the Poisson spheres, are studied.