On the Invariant Separated Variables

An integrable Hamiltonian system on a Poisson manifold consists of a Lagrangian foliation $\mathscr{F}$ and a Hamilton function $H$. The invariant separated variables are independent on values of integrals of motion and Casimir functions. It means that they are invariant with respect to abelian group of symplectic diffeomorphisms of $\mathscr{F}$ and belong to the invariant intersection of all the subfoliations of $\mathscr{F}$. In this paper we show that for many known integrable systems this invariance property allows us to calculate their separated variables explicitly.