Integrable homogeneous dynamical systems with dissipation on the tangent bundles of smooth finite-dimensional manifolds. I. Equations of geodesics on the tangent bundle of a smooth $n$-dimensional manifold

In many problems of dynamics, systems arise whose position spaces are four-dimensional manifolds. Naturally, the phase spaces of such systems are the tangent bundles of the corresponding manifolds. Dynamical systems considered have variable dissipation, and the complete list of first integrals consists of transcendental functions expressed in terms of finite combinations of elementary functions. In this paper, we prove the integrability of more general classes of homogeneous dynamical systems with variable dissipation on tangent bundles of four-dimensional manifolds.