As I understand it, for a mechanical system, each symmetry leads to a constant of motion. For integrable systems, the number of constants of motion equals the number of degrees of freedom.
So take 2D square and circular billiards tables. What are their constants of motion? For the circular table, it looks like angular momentum is a constant of motion. For the square table, it looks like $\left|v_x\right|$ and $\left|v_y\right|$ (the magnitudes of the x and y components of the velocities) are constant.
Are those indeed constants of motion? If so, are there more (or other ways to view the constants of motion)? (It seems like there should be a second for the circular table.)