Abstract <p>We present new cases of integrable dynamical systems of any odd order that are homogeneous in terms of some of their variables and in which a system on the cotangent bundle of an even-dimensional manifold can be distinguished. In this case, the force field (shift generator in the system) 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.</p>