The gyrostat motion around the inertia center in the semi-Euclidean space

The inertial motion of a gyrostat in a semi-Euclidean space with given index and defect is studied. A gyrostat with a constant gyrostatic moment moves so that its carrier rotates around a fixed center of inertia. Criteria for the existence of regular motions are obtained as a condition for the presence of axial structural-kinetic symmetry of the gyrostat. The properties of nutational, precessional, vibrational-rotational motions are studied and their description in configuration and phase spaces is given. The quadrature dependences of the gyrostat motion parameters in elliptic functions of time are determined. Parametric equations of hodographs of angular velocity and kinetic momentum vectors are found. The study was carried out for the case of the gyrostat angular momentum eigenvector.