A system is completely integrable (in the Liouville sense) if there exist $n$ Poisson commuting first integrals. The Liouville-Arnold theorem, anyway, requires additional topological conditions to find a transformation which leads to action-angle coordinates and, in these set of variables, the Hamilton-Jacobi equation associated to the system is completely separable so that it is solvable by quadratures.
What I would like to understand is if the additional requirement of the Liouville-Arnold theorem (the existence of a compact level set of the first integrals in which the first integrals are mutually independent) means, in practice, that a problem with an unbounded orbit is not treatable with this technique (for example the Kepler problem with parabolic trajectory).
If so, what is there a general approach to systems that have $n$ first integrals but do not fulfill the other requirements of Arnold-Liouville theorem? Are they still integrable in some way?
Let $M= \{ (p,q) \in \mathbb{R}^{n} \times \mathbb{R}^n \}$ ($p$ denotes the position variables and $q$ the corresponding momenta variables). Assume that $f_1, \cdots f_n$ are $n$ commuting first integrals then you get that $M_{z_1, \cdots, z_n} := \{ (p,q) \in M \; : \; f_1(p,q)=z_1, \cdots , f_n(p,q)=z_n \} $ with $z_i \in \mathbb{R}$ is a Lagrangian submanifold.
Observe that if the compactness and connectedness condition is satisfied then there exist action angle variables which means that the motion lies on an $n$-dimensional torus (which is a compact object).
The compactness condition is equivalent to that a position variable, $p_k$, or a momentum variable, $q_j$, cannot become unbounded for fixed $z_i$. Consequently, if the compactness condition is not satisfied there is no way you can expect to find action angle variables since action angle variable imply that the motion lies on a torus which is a compact object.