Complete commutative subalgebras in polynomial Poisson algebras: a proof of the Mischenko--Fomenko conjecture

The Mishchenko-Fomenko conjecture says that for each real or complex finite-dimensional Lie algebra $\text{goth}g$ there exists a complete set of commuting polynomials on its dual space $\text{goth}{g}^{\ast }$. In terms of the theory of integrable Hamiltonian systems this means that the dual space $\text{goth}{g}^{\ast }$ endowed with the standard Lie-Poisson bracket admits polynomial integrable Hamiltonian systems. Recently this conjecture has been proved by S.T. Sadetov. Following his idea, we give an explicit geometric construction for commuting polynomials on $\text{goth}{g}^{\ast }$ and consider some examples.