A new solution of Kirchhoff's equations in the case of a linear invariant relation

The problem of integrating Kirchhoff's differential equations [1] when they allow of a linear invariant relation with respect to the main variables - the components of the angular momentum of a gyrostat and the unit vector of the axis of symmetry of the force field, is considered. The initial system of equations is reduced to a second-order system using first integrals of the equations. Under certain conditions, imposed on the parameters characterizing the geometry of the gyrostat masses and the potential and gyroscopic forces, the integrating factor of the reduced equations is obtained. The solution of Kirchhoff's equations obtained contains four arbitrary constants and is determined for more general assumptions compared with existing solutions [2-4].