Suppose I have an integrable Hamiltonian system $H(q_{1}, p_{1},..., q_{n}, p_{n})$, with first integrals $F_{1} = H, F_{2},..., F_{n}$. Excluding certain singular level sets (i.e. separatrices), one may convert to action-angle variables $(I, \varphi)$, (at least locally in the complement of singular level sets), and write $H(I)$.
My question is, how exactly are actions $I$ related to first integrals $F$? We know that generally, $I_{i} =I_{i}(F_{1},..., F_{n})$, for $i=1,..,n$, i.e. $I$'s are functions of $F$'s. Can we say anything further about the specific form of these functions?
More specifically, for what type of Hamiltonians can we take $I_{i} = F_{i}$?
Edit. My motivation comes from the following Hamiltonian: $$H = H(q_{1}, p_{1},..., q_{j}, p_{j}, F_{k}(q_{j+1}, p_{j+1}), q_{j+2}, p_{j+2},..., q_{n}, p_{n})$$
where $F_{k}$ is one of the first integrals, that depends only on $(q_{j+1}, p_{j+1})$. Can one, in this case, take $I_{k} = F_{k}$?
As a further motivation, consider a one-dimensional oscillator $H = p^{2} + q^{2}$; here one may simply take $I = H$ (up to an adjustment of a constant factor). How does this generalise to the type of Hamiltonian given above?
