defined especially remarkable motions as such solutions of the corresponding Euler–Poisson equations for which one of the separated Kowalevski variables remains constant. He pointed out four classes of such motions depending on various constraints on the integral constants leading to multiple roots of the Kowalevski polynomial involved in the separated equations. The corresponding solutions are expressed as elliptic functions of time. More thorough analytical and geometrical study of these motions was given by A.F.Ipatov [@Ipat]. Later it appeared that the Appelrot classes play the main role in the topological analysis of the Kowalevski case; the union of all trajectories of the especially remarkable motions in the phase space form a critical set of the integral map determining bifurcations of the integral manifolds [@KhPMM] (the rough topology of the Liouville foliation) and exact topological invariants of the problem [@BRF].

The Appelrot classes received many generalizations in various directions. In [@PVLect] a new integrable case was found for a heavy dynamically symmetric gyrostat (a rigid body with a rotor characterized by constant gyrostatic momentum ${\boldsymbol \lambda}$); this case includes the Bobylev–Steklov case and the 4th Appelrot class as partial problems under some restrictions on the parameters. Another two exact solutions for a 