  Let G be an algebraic group over a field F of characteristic zero, with Lie
algebra g=Lie(G). The dual space g^* equipped with the Kirillov bracket is a
Poisson variety and each irreducible G-invariant subvariety X\subset g^*
carries the induced Poisson structure. We prove that there is a family of
algebraically independent polynomial functions {f_1,...f_l} on X, which
pairwise commute with respect to the Poisson bracket and such that l=(dim
X+tr.deg F(X)^G)/2. We also discuss several applications of this result to
complete integrability of Hamiltonian systems on symplectic Hamiltonian
G-varieties of corank zero and 2.
