Classification of integrable cases in the dynamics of a four-dimensional rigid body in a nonconservative field in the presence of a tracking force

This paper is a survey of integrable cases in the dynamics of a four-dimensional rigid body under the action of a nonconservative force field. We review both new results and results obtained earlier. The problems examined are described by dynamical systems with so-called variable dissipation with zero mean.
The problem of a search for complete sets of transcendental first integrals of systems with dissipation is quite current; a large number of works are devoted to it. We introduce a new class of dynamical systems that have a periodic coordinate. Due to the existence of a nontrivial symmetry group of such systems, we can prove that these systems possess variable dissipation with zero mean, which means that on the average for a period with respect to the periodic coordinate, the dissipation in the system is equal to zero, although in various domains of the phase space, either energy pumping or dissipation can occur. Based on the results obtained, we analyze dynamical systems that appear in the dynamics of a four-dimensional rigid body and obtain a series of new cases of complete integrability of the equations of motion in transcendental functions that can be expressed through a finite combination of elementary functions.