On the conditions of existence of Steklov-Bobylev partial integrals

The paper describes the results of investigation of one earlier known relationship regarding the inertia moments of the rigid body B = 2 A for the case of existence of the Steklov-Bobylev partial integrals. Obtaining of the relation for inertia moments relies on the possibility of constructing additional partial integrals and some real solutions of the differential equations of motion. Obtaining the integrals indicated is based on obtaining the most general nontrivial solutions of equations of motion for a body in case vanishing of the right-hand side of the differential equation for q = 0 . In one particular case of existence of such solutions, the requirement of A ≠ B , which is consistent with the known relationship given above, has been obtained. Indeed, the set of partial Steklov-Bobylev integrals may be derived from the condition B ≠ C. The relationship for the number of possible additional first integrals depending on the values of the body’s inertia moments has been obtained. The largest number of additional integrals, while including two Steklov-Bobylev integrals, is possible for all various inertia moments. The same number of additional integrals may exists for a dynamically symmetric rigid body having only two equal inertia moments, whose mass center is displaced with respect to the origin on the symmetry axis. In this case, the general Lagrange integral participates. As far as other cases of symmetry with only two equal inertia moments are concerned, for which displacement of the mass center with respect to the origin is realized not along the symmetry axis, only one of the Steklov-Bobylev integrals is admissible. In the case of sphere, the partial Steklov-Bobylev integrals do not exist, and only the general Lagrange integral may additionally participate.