  For one 3-body and two 5-body planar choreographies on the same algebraic
Lemniscate by Bernoulli we found explicitly a maximal possible set of
(particular) Liouville integrals, 7 and 15, respectively, (including the total
angular momentum), which Poisson commute with the corresponding Hamiltonian
along the trajectory. Thus, these choreographies are particularly maximally
superintegrable. It is conjectured that the total number of (particular)
Liouville integrals is maximal possible for any odd number of bodies $(2n+1)$
moving choreographically (without collisions) along given algebraic Lemniscate,
thus, the corresponding trajectory is particularly, maximally superintegrable.
Some of these Liouville integrals are presented explicitly. The limit $n \rar
\infty$ is studied: it is predicted that one-dimensional liquid with
nearest-neighbor interactions occurs, it moves along algebraic Lemniscate and
it is characterized by infinitely-many constants of motion.
