New classes of particular solutions to one problem on gyrostat motion

The paper studies the existence of two new classes of polynomial solutions to differential equations related to the problem of the gyrostat motion with a fixed point in the magnetic field, taking into account the Barnett-London effect. A common feature of the structure of these classes is that the functions that set the invariance relations for the unit vector components of the symmetry axis of the active force fields are either rational functions of the first component of the specified vector or of the auxiliary variable. Three new particular solutions to the polynomial classes under consideration are constructed. These solutions are described by the functions obtained by the inversion of hyperelliptic integrals. It has been proved that another constructed solution of the polynomial structures under study, for which the movement of the gyrostat has the property of precession, is a particular case of a known solution.