Integrable homogeneous dynamical systems with dissipation on the tangent bundles of smooth finite-dimensional manifolds. III. Equations of motion on the tangent bundle of an $n$-dimensional manifold in a force field with variable dissipation

This paper is the conclusion of the work on the integrability of general classes of homogeneous dynamical systems with variable dissipation on the tangent bundles of $n$-dimensional manifolds. The first part of the paper is: Integrable homogeneous dynamical systems with dissipation on the tangent bundles of smooth finite-dimensional manifolds. I. Equations of geodesics on the tangent bundle of a smooth $n$-dimensional manifold// Itogi Nauki Tekhn. Sovr. Mat. Prilozh. Temat. Obzory. — 2022. — V. xxx. — P. xx–xx. The second part of the paper is: Integrable homogeneous dynamical systems with dissipation on the tangent bundles of smooth finite-dimensional manifolds. II. Equations of motion on the tangent bundle of an $n$-dimensional manifold in a potential force field// Itogi Nauki Tekhn. Sovr. Mat. Prilozh. Temat. Obzory. — 2022. — V. xxx. — P. xx–xx.