A new case of integrability in the dynamics of a 4D-rigid body in a nonconservative field under the assumption of linear damping

The dynamic part of equations of motion of a dynamically symmetric 4D-rigid body, where the force field is accumulated on that part of the body surface that is shaped as a three-dimensional ball, is investigated. In this case, the tensor of the angular velocity of such a body is six-dimensional, while the velocity of the center of mass is four-dimensional. Under certain conditions, the complete list of first integrals, which are expressed through elementary functions, is obtained.