Invariants of Fifth-Order Homogeneous Systems with Dissipation

New cases of integrable fifth-order dynamical systems that are homogeneous with respect to some of the variables are obtained, in which a system on the tangent bundle of a two-dimensional manifold can be distinguished. In this case, the force field is divided into an internal (conservative) and an external one, which has dissipation of different signs. The external field is introduced using some unimodular transformation and generalizes previously considered fields. Complete sets of both first integrals and invariant differential forms are given.