I hesitate to say you were wrong, but in that paper it appears as though they weren't "clamping" angular momentum, just the vis viva energy. On a different question, I don't really follow the requirements of degrees of freedom vis-a-vis constants of motion for an integrable mechanical system. Unfortunately I don't even know how to begin formulating a question in this area. It would come out sounding like total nonsense, even to me. For example "To improve the integrability of a system, would you prefer to reduce the number of degrees of freedom, increase the number of constants of motion, or the opposite of either one of those?" As far as I know, that question is complete gibberish.