Systems with dissipation with a finite number of degrees of freedom: analysis and integrability. I. Primordial problem from dynamics of a multidimensional rigid body in a nonconservative field of forces

This paper is the first part of a survey on the integrability of systems with a large number $n$ of degrees of freedom. The review consists of three parts. In this first part, the primordial problem from the dynamics of a multidimensional rigid body placed in a nonconservative force field is described in detail. In the second and third parts, which will be published in the next issues, we consider more general dynamical systems on the tangent bundles to the $n$-dimensional sphere and other smooth manifolds of a sufficiently wide class. Theorems on sufficient conditions for the integrability of the considered dynamical systems in the class of transcendental functions are proved.