Abstract
We analyse the phase flow evolution of the torque free asymmetric gyrostat motion. The gyrostat consists of a triaxial rigid body and a symmetric rotor spinning around one of the principal axis of inertia of the gyrostat. The problem is converted into a two parametric quadratic Hamiltonian with the phase space on the sphere. As the parameters evolve, the appearance - disappearance of centres and saddle points is originated by a sequence of pitchfork bifurcations. When the gyrostat is axial symmetric, there are motions of the rotor that break the degeneracy through an oyster bifurcation while other motions simply shift the degeneracy along a minor circle.
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