Topological atlases for some integrable problems of rigid body dynamics

The report presents some results of a study of the phase topology of two integrable Hamiltonian systems. The first one refers to the general case of integrability on the Lie algebra $so(4)$, found by M. Adler and P. van Moerbeke. The second system is the integrable Hamiltonian system with three degrees of freedom, which describes the dynamics of the so-called generalized two-field gyrostat. Integrability of the second system was proved by V.V. Sokolov and A.V. Tsiganov. In the report for both systems we explicitly give first integrals which are the coefficients of the spectral curve. This enables us upon the inspection of the curves singularities to find the bifurcation diagram of the momentum map (the image of the critical points of the momentum map). For both systems we construct the bifurcation diagrams and explore the bifurcations of Liouville tori.