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

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