Separation of variables in the generalized 4th Appelrot class

Abstract

We consider an analogue of the 4th Appelrot class of motions of the Kowalevski top for the case of two constant force fields. The trajectories of this family fill a four-dimensional surface $$\mathfrak{O}$$ in the six-dimensional phase space. The constants of the three first integrals in involution restricted to this surface fill one of the sheets of the bifurcation diagram in ℝ3. We point out a pair of partial integrals to obtain explicit parametric equations of this sheet. The induced system on $$\mathfrak{O}$$ is shown to be Hamiltonian with two degrees of freedom having a thin set of points where the induced symplectic structure degenerates. The region of existence of motions in terms of the integral constants is found. We provide the separation of variables on $$\mathfrak{O}$$ and algebraic formulae for the initial phase variables.