gyrostat with the Kowalevski type inertia tensor were found in [@EIPVHDan; @PVMtt71]. These solutions cover the 2nd and the 3rd Appelrot classes as ${\boldsymbol \lambda}$ tends to zero. After the Kowalevski type heavy gyrostat was proved to be completely integrable [@Yeh], it appeared that the motions pointed out in [@PVLect; @EIPVHDan; @PVMtt71] form a critical set of the Kowalevski–Yehia gyrostat [@Ryab1]. Other analogues of the Appelrot classes are associated with the problem of the rigid body motion in a *double force field* (sometimes referred to as *two constant fields*). This field could be, for example, a superposition of the gravity field and the constant magnetic field affecting a heavy magnetized rigid body with a fixed point. The equations of motion were given by O.I.Bogoyavlensky in [@Bogo] and presented as the Euler equation on the Lie co-algebra $e(3,2)^*$. In the same paper a partial integrable case (the subsystem with two degrees of freedom) was found generalizing the 1st Appelrot class. More general system including a double field and a non-zero gyrostatic momentum was proved integrable in [@ReySem]. In this system, the generalizations of the 2nd, 3rd and 4th Appelrot classes with ${\boldsymbol \lambda}=0$ were discovered in [@Kh32; @KhRCD]. For non-zero gyrostatic 