ИСТИНА

I.V.~Komarov in his paper [1] 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 $\textrm{so}(3,1)-\textrm{e}(3)-\textrm{so}(4)$ with real parameter $\ae \in \mathbf{R}$. The Kovalevskaya top was realized as a system on $\textrm{e}(3)$ for $\ae =0$. The following bracket on $\mathbf{R}^6$ depends on $\ae$: \begin{equation} \label{Eq:poisson_bracket_so4} \{J_i, J_j\} = \varepsilon_{ijk}J_k, \quad \{J_i, x_j\} = \varepsilon_{ijk}x_k, \quad \{x_i, x_j\} = \ae \varepsilon_{ijk}J_k, \end{equation} where $\varepsilon_{ijk}$ is the sign of permutation $\{123\} \rightarrow \{ijk\}$, $\ae \in \mathbf{R}$. When $\ae >0, \ae = 0, \ae <0$ this bracket coincides with the Lie--Poisson brackets for the Lie algebras $\textrm{so}(4), \textrm{e}(3), \textrm{so}(3, 1)$ respectively. These brackets have common Casimir functions \begin{equation} f_1 = (x_1^2 + x_2^2 +x_3^2) + \ae (J_1^2 + J_2^2 +J_3^2), \quad f_2 = x_1 J_1 +x_2 J_2 +x_3 J_3. \end{equation} The Hamiltonian and an additional integral of these systems are equal to \begin{equation}H = J_1^2 +J_2^2 +2 J_3^2 + 2 c_1 x_1, \end{equation} \begin{equation} K = (J_1^2 - J_2^2 - 2 c_1 x_1 + \ae c_1^2)^2 + (2 J_1 J_2 - 2 c_1 x_2)^2. \end{equation} We consider three-dimensional submanifolds which are common level surfaces of first integrals of the system. Topology type of them (class of diffeomorphic manifolds) can be determined by several ways. One of them is analysis of Fomenko-Zieschang invariants. They were calculated in [2] for the classical Kovalevskaya case and by speaker for the case of the Lie algebra $\mathrm{so}(4)$. This work was supported by the Russian Science Foundation grant (project No.17-11-01303). \medskip \centerline{\textsl{REFERENCES}} \smallskip \small \lit{1}{Komarov I.\,V.,} {``Kowalewski basis for the hydrogen atom,'' Theoret. and Math. Phys., \textbf{47}, No. 1, 320--324 (1981).} \lit{2}{Bolsinov A.\,V., Richter P., Fomenko A.\,T.,} {``The method of loop molecules and the topology of the Kovalevskaya top,'' Sb. Math., \textbf{191}, No. 2, 3--42 (2000).}