Linear invariant relation for equations of motion of a gyrostat with variable gyrostatic moment in the magnetic field

The author studies equations of motion of a heavy gyrostat in the magnetic field in the presence of Barnet-London effect. Existence conditions are obtained for one invariant relation, linear in the state variables (they are the components of the angular velocity vector and the components of the vertical unit vector in the body axis). It is shown that, under these conditions, the axis determined by the linear invariant relation is orthogonal to the circular cross-section of the gyratory ellipsoid. Asymptotically steady motions of the gyrostat relative to the vertical line are obtained in the case, when the gyrostatic moment depends on time.