It is found explicitly 5 Liouville integrals in addition to total angular momentum which Poisson commute with Hamiltonian of 3-body Newtonian Gravity in ${ \mathbb R}^3$ along the Remarkable Figure-8-shape trajectory discovered by Moore-Chenciner-Montgomery. It is shown they become constants of motion along this trajectory. Hence, 3-body choreographic motion on Figure-8-shape trajectory in ${ \mathbb R}^3$ Newtonian gravity (Moore, 1993), as well as in ${ \mathbb R}^2$ modified Newtonian gravity by Fujiwara et al, 2003, is maximally superintegrable. It is conjectured that any 3-body potential theory which admit Figure-8-shape choreographic motion is superintegrable along the trajectory.
Categories: physics.class-ph hep-th math-ph math.DS math.MP
