The absence of an additional first integral in a rigid-body dynamics problem

The paper deals with a classical problem in nonholonomic mechanics concerning the motion of a heavy rigid body about a fixed point with a zero projection of the angular velocity vector onto a direction that is stationary in rotating coordinates. It is assumed that the direction corresponding to the zero projection of the angular velocity coincides with one of the main axes of inertia. A particular case is analyzed where the center of weight is located on this axis. Theorems are proved on the absence of an additional first integral in the system investigated.