Topological features of the Sokolov integrable case on the Lie algebra $\mathrm{so}(3,1)$

The integrable Sokolov case on $\mathrm{so}(3,1)^{\star}$ is investigated. This is a Hamiltonian system with two degrees of freedom, in which the Hamiltonian and the additional integral are homogeneous polynomials of degrees 2 and 4, respectively. It is an interesting feature of this system that connected components of common level surfaces of the Hamiltonian and the additional integral turn out to be noncompact. The critical points of the moment map and their indices are found, the bifurcation diagram is constructed, and the topology of noncompact level surfaces is determined, that is, the closures of solutions of the Sokolov system on $\mathrm{so}(3,1)$ are described.
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