Integrable homogeneous dynamical systems with dissipation on the tangent bundle of a two-dimensional manifold

In many problems of dynamics, position spaces of systems considered are two-dimensional manifolds; the phase spaces of these systems are the tangent bundles of the corresponding manifolds. For example, the study of a spatial pendulum on a spherical hinge in a flow of a medium leads to a dynamical system on the tangent bundle of the two-dimensional sphere; in this case, a metric of a special form on the sphere is induced by an additional symmetry group. In such cases, dynamical systems have variable dissipation, and the complete list of first integrals consists of transcendental functions expressed as finite combinations of elementary functions. For problems on the motion of a point on a two-dimensional surface, the metric on the surface is induced by the Euclidean metric of the ambient space. In this paper, we prove the integrability of more general classes of homogeneous dynamical systems on tangent bundles of two-dimensional manifolds that involve force fields with variable dissipation.