Some tensor invariants of geodesic, potential, and dissipative systems on the tangent bundles of three-dimensional manifolds

In this paper, we present tensor invariants (differential forms) for homogeneous dynamical systems on the tangent bundles of smooth three-dimensional manifolds and demonstrate the connection between the presence of these invariants and the existence of a complete set of first integrals, which is necessary for integrating geodesic, potential, and dissipative systems.