On the qualitative analysis of a family of differential equations with first integrals of degree more then 2

We study a family of differential equations that arose as a result of a generalization of the classical integrable cases in rigid body dynamics. The system under study admits polynomial first integrals of the 4th and 6th degrees. Under appropriate constraints on the parameters of the family, the differential equations are interpreted as those of motion of a rigid body in a central force field and an ideal fluid, as well as the equations of motion of an electrically charged body. The qualitative analysis of the equations is done. We find special invariant sets of various dimensions and investigate their Lyapunov stability. For the analysis of the problem, generalizations of the Routh-Lyapunov method and software tools of computer algebra are applied.