momentum, analogues of the Appelrot classes were found in [@KhHMJ]. In the general case these analogues also form a critical set of the integrable map; the critical trajectories are organized in almost everywhere Hamiltonian systems with less than three degrees of freedom (so-called critical subsystems). The knowledge of critical subsystems made it possible to obtain many analytical results (e.g, new separation of variables [@KhSavUMBeng; @KhRCD09], explicit descriptions of periodic motions [@KhJPA; @KhIISmir40]) and to fulfill a number of topological investigations [@KhSavUMBeng; @KhRCD09; @KhND11]. On this basis, the complete description of the topology of the Kowalevski top in a double field in terms of net topological invariants was given [@RyabKhMatsbEn; @KhRyabDME].

In 2002, new integrable problems for the Kowalevski type gyrostat were found by V.V.Sokolov for the case of one axially symmetric field [@SokVV] and by V.V.Sokolov and A.V.Tsiganov for the case of a double field [@SokTsiE]. The last one also generalizes the Kowalevski type integrable gyrostat found by A.G.Reyman and M.A.Semenov-Tian-Shansky [@ReySem]. The investigation of the phase topology of the Sokolov–Tsiganov case was started in [@RyabTMF].

In the present paper we obtain the generalized Appelrot classes for the case found in [@SokTsiE] and establish some properties of the appearing invariant 