On invariant relation of the motion equations of a heavy gyrostat when the center of inertia belongs to the principal plane

The paper concerns rotation of a heavy gyrostat with a fixed point under the assumption that the gyrostatic momentum direction is invariant in the rotating frame. In addition, this directing vector and the position vector of the center of inertia belong to the principal plane. The existence conditions for solutions of the motion equations are studied in the case of one linear in angular velocity invariant relation. The equivalent of known Kharlamova’s particular solution is obtained.