A system with a $2n$-dimensional phase space is Liouville-integrable if it admits $n$ independent first intgrals in involution.

Here integrable means that you can, in some way, solve the equations of motion by quadratures.

The Liouville-Arnol'd theorem states that a Liouville-integrable system admits a canonical transformation to action-angle coordinates, provided that it respects some other topological conditions.

These are that the level set of the first integrals must be compact and connected. My question is: is this condition very restrictive in the usual case? And does it imply that the orbit is quasi-periodic under those conditions?

I wonder if a problem like the Kepler problem with an open orbit (when the energy is greater than zero) is treatable with the Arnold-Liouville method.