classical mechanics - Is a system Liouville integrable if and only if its Hamilton-Jacobi equation is separable?

I am asked to show that, a system is completely integrable Liouville if and only if its Hamilton-Jacobi equation is completely separable. I get the idea and understand that is very related to the Action-Angle coordinates and one of Liouville's theorems. Two texts have been of help: Jose & Saletan and Arnol'd.

I see how separability leads to AA coordinates and thus to an integrable solution of the system, but why is it an "if and only if" relationship? Why can't there exist systems with non-separable HJ, but reducible to integrals?