Geometric and dynamical invariants of integrable Hamiltonian and dissipative systems

This paper presents results referred to geometric invariant theory of completely integrable Hamiltonian systems and also to the classification of integrable cases of low-dimensional and high-dimensional rigid body dynamics in a nonconservative force field. The latter problems are described by dynamical systems with variable dissipation. The first part of the work is the base the doctorial dissertation of V. V. Trofimov (1953–2003), which was in parts already published. However, in the present entire form, it was not appeared, and we decided to fill in this gap. The second part is a development of the results presented in the doctorial dissertation of M. V. Shamolin, and it was not appeared in the present variant. These two parts well complement one another, which initiated this work (its sketches already appeared in 1997).