Completely integrable Hamiltonian systems on semidirect sums of Lie algebras

The complete integrability of Hamiltonian systems arising on Lie algebras which have the form of a direct sum is investigated. For algebras in these classes Sadetov's method takes a simpler form: the isomorphism between the algebra arising at the second step of Sadetov's approach and the stationary subalgebra of a generic element can be written out explicitly. The explicit form of this isomorphism is presented, as well as explicit formulae for polynomials in complete systems for the algebras $\operatorname{so}(n)+(\mathbb{R}^n)_k$, $\operatorname{su}(n)+(\mathbb{C}^n)_k$ and $\mathrm u(n)+(\mathbb{C}^n)_k$. For the algebras $\operatorname{so}(n)+\mathbb{R}^n$ the degrees of the resulting polynomial functions are analysed.
Bibliography: 15 titles.