The study of (maximally) superintegrable systems, i.e., Hamiltonian systems of \(n\) degrees of freedom possessing the maximal number \(2n-1\) of functionally independent integrals of motion, has a long standing history since the original work of Nekhoroshev and others in the early 70’s, and attracting much attention from the fundamental non-commutativity theorem due to Mishchenko and Fomenko, which generalizes Arnold-Liouville integrability. Since then there has been a number of studies of many important and physically interesting theories, which have been shown to be superintegrable both in classical and quantum mechanics.
In the paper under review, the authors raise the question whether recently discovered generalization of Calogero-Moser systems associated with cyclic quivers is too superintegrable. There are several cases investigated in the paper – spinless, spin, spin-harmonic, each case being illustrated with several helpful examples and presented with complete explicit technical constructions.
The main result of the paper is that all these cases are superintegrable, and the authors show that ultimately, for all the considered cases, the superintegrability follows from two basic theorems which the authors also formulate. The last and very useful and helpful section of the paper is devoted entirely to detailed proofs of all statements and theorems used in the main text.