Topological features of the Sokolov integrable case on the Lie algebra $\mathrm{e}(3)$

The Sokolov integrable case on $\mathrm{e}(3)^{\star}$ is investigated. This is a Hamiltonian system with $2$ degrees of freedom in which the Hamiltonian and the additional integral are homogeneous polynomials having degree $2$ and $4$, respectively. This system is of interest because connected joint level surfaces of the Hamiltonian and the additional integral are noncompact. The critical points of the moment map and their indices are found, the bifurcation diagram is constructed and the Liouville foliation of the system is described. The Hamiltonian vector fields corresponding to the Hamiltonian and the additional integral are proved to be complete.
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