ИСТИНА

I.V.~Komarov showed that the Kovalevskaya integrable case in rigid body dynamics can be included in a one parameter family of integrable Hamiltonian systems on the pencil of Lie algebras so(3,1)-e(3)-so(4) with real parameter $\ae$. The Kovalevskaya top was realized as a system on e(3) for $\ae =0$. When $\ae >0, \ae = 0, \ae <0$ some Poisson bracket coincides with the Lie--Poisson brackets for the Lie algebras so(4), e(3), so(3, 1) respectively. The case of $\ae >0$ was investigated by I. Kozlov and V. Kibkalo. The case of $\ae <0$ is closely related to Sokolov integrable case. Phase topology of that system was investigated by P. Ryabov. Topology of isoenergy manifolds and Fomenko graphs for them were determined for Kovalevskaya case for $\ae <0$. We consider isoenergy manifolds $Q^3_{a, b, h}$ and calculate Fomenko--Zieschang invariant for the Liouville foliations on them. These classes of Liouville equivalence are compared with one for other integrable systems investigated earlier and billiards in confocal quadrics. Some of them are equivalent, so these systems has the same structure of closure of trajectories on several energy levels. This work was supported by the Russian Science Foundation grant (project No. 17-11-01303).