I am currently studying the Arnold-Liouville theorem, more precisely the construction of action-angle coordinates. I am following mainly the books "Physics for mathematicians I: Mechanics" by Spivak and "Mathematical Methods of Classical Mechanics" by Arnold.
On the first part of the theorem one proves that the connected components of the level sets of the map $F=(F_1,...,F_n)$ of the integrals in involution are diffeomorphic to $\mathbb{T}^k \times \mathbb{R}^{n-k}$, assuming that the fields $X^{F_i}$ are all complete.
After that, action-angle variables are constructed, but just for the level sets that are compact, i.e., those that are indeed tori. The key part of the proof relies on the fact that, around each torus a "product neighbourhood" can be built, this is a neighbourhood $U$ of the level set that is diffeomorphic to $\mathbb{R}^n\times \mathbb{T}^n$.
My question is, since "angle" variables with trivial flow can be defined for the general case of non-compact level sets, can you get action-angle variables in the general case? Can you get a "product neighbourhood" diffeomorphic to $\mathbb{R}^n \times (\mathbb{T}^k \times \mathbb{R}^{n-k})$ in this general case?
If the answer is not in general, does this mean that in general in those level sets the system cannot be integrated by quadratures, or that you cannot find Darboux coordinates which integrate the system by quadratures globally?
