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 →∞ 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.

中文翻译：

---

(2n + 1)-body choreographies, n = 1,2,3,...,∞ 在 Bernoulli 的代数双线上的超可积性（经典力学的反问题）

对于 Bernoulli 在同一代数双线上的一个 3 体和两个 5 体平面编排，我们明确地发现了一组最大可能的（特定）刘维尔积分，分别为 7 和 15（包括总角动量），Poisson 对沿轨迹具有相应的哈密顿量。因此，这些编排特别是最大可超积的。推测对于任何奇数个物体，（特定）刘维尔积分的总数是最大的(2n + 1)沿着给定的代数双纽节编排地移动（没有碰撞），因此，相应的轨迹特别是最大可超积的。其中一些刘维尔积分是明确提出的。限制n →∞研究：预测会发生具有最近邻相互作用的一维液体，它沿着代数双纽线运动，并且具有无限多个运动常数的特征。