Partial linear integrals of the Poincaré–Zhukovskii equations (the general case)

The existence conditions for a linear invariant relation of the Poincaré–Zhukovskii equations in the general case when the matrix of the cross terms of the Hamiltonian can be asymmetric are obtained. A new scalar form of the equations is indicated, and they are reduced to the Riccati equation in the case of motion with a linear invariant relation. A particular solution of the Riccati equation, which defines a three-parameter family of periodic solutions of the Poincaré–Zhukovskii equations, is presented. A four-parameter family of solutions of the Poincaré–Zhukovskii equations, each of which exponentially rapidly approaches a corresponding periodic solution with time, is constructed. The conditions for precessional motion with a linear invariant relation are found.