Invariant volume forms of variable dissipation systems with three degrees of freedom

Tensor invariants (differential forms) for homogeneous dynamical systems on tangent bundles of smooth three-dimensional manifolds are presented. The connection between the presence of these invariants and the full set of first integrals necessary for the integration of geodesic, potential, and dissipative systems is shown. The introduced force fields make the considered systems dissipative with dissipation of different signs and generalize previously considered fields.