Consider a hamiltonian toric acion on a connected real symplectic manifold of dimension 2n. The dimension of the torus, which we denote by $k$, may be less than $n$. The generators of the action will be denoted by $F_1,...,F_k$, they Poisson commute and their differentials are linearly independent at one point, and, therefore, at almost every point of the manifold. The Hamiltonian flow of each of these functions is periodic with period $1$.
Then, for any function $H$ of $k$ variables the Hamiltonian flow generated by $H(F_1,..., F_k)$ is Liouville integrable. More precisely, the following statement holds:
Theorem. One can find functions $F_{k+1}, ... , F_{n}$ such that all the functions $F_1,..., F_n$ Poisson commute and their differentials are linearly independent at almost every point of the manifold.
Question. Do you know a reference where the Theorem above is proved or at least formulated.
Background. In the case $k=0$ Theorem is reduced to the following statement: on each symplectic manifold there exists a Liouville-integrable system. This statement is proposition 3.3. of the book "Symplectic geometry" of Fomenko (english version; in the russian version of the book the statement is Proposition 1 in §5.1). And in fact the proof of Fomenko can be generalizeed for all $k$, after some work.
