On two linear invariant relations of gyrostat motion equations with a variable gyrostatic momentum under the action of potential and gyroscopic forces

The problem of gyrostat motion under the action of potential and gyroscopic forces in the case of variable gyrostatic momentum is considered. Existence conditions for two linear in all variables invariant relations of the motion equations are defined. Time dependence of the gyrostatic momentum value and the set of parameter conditions that characterize the new solution of the generalized Kirchhoff equations are found.