Integrability of the problem of motion of a body with a fixed point in a flow of particles

The problem of the motion, in the free molecular flow of particles, of a rigid body with a fixed point is considered. The molecular flow is assumed to be sufficiently sparse, there is no interaction between the particles. Based on the approach proposed by V.V. Beletsky, an expression is obtained for the moment of forces acting on a body with a fixed point. It is shown that the equations of motion of a body are similar to the classical Euler-Poisson equations of motion of a heavy rigid body with a fixed point and are presented in the form of classical Euler-Poisson equations in the case when the surface of a body is a sphere. The existence of the first integrals is discussed. Constraints on the system parameters are obtained under which there are integrable cases corresponding to the classical Euler-Poinsot, Lagrange and Hess cases of integrability of the equations of motion of a heavy rigid body with a fixed point. The case when the surface of the body is an ellipsoid is considered. Using the methods developed in the works of V.V. Kozlov, proved the absence of an integrable case in this problem, similar to the Kovalevskaya case.