Systems with dissipation with a finite number of degrees of freedom: analysis and integrability. III. Systems on the tangent bundles of smooth $n$-dimensional manifolds

This paper is the third part of a survey on the integrability of systems with a large number $n$ of degrees of freedom (the first part: Itogi Nauki Tekhn. Sovr. Mat. Prilozh. Temat. Obzory, 211 (2022), pp. 41–74; the second part: Itogi Nauki Tekhn. Sovr. Mat. Prilozh. Temat. Obzory, 212 (2022), pp. 139–148). The review consists of three parts. In the first part, the primordial problem from the dynamics of a multidimensional rigid body placed in a nonconservative force field is described in detail. The second part is devoted to more general dynamical systems on the tangent bundles to the $n$-dimensional sphere. In this third part, we discuss dynamical systems on the tangent bundles to 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.