Superintegrable Extensions of Superintegrable Systems

TL;DR

This paper presents a systematic procedure to extend known superintegrable systems into new ones, successfully applied to systems on Euclidean and spherical spaces, including notable models like TTW and Calogero systems.

Contribution

The paper introduces a systematic method for extending superintegrable systems, demonstrated on various classical models and manifolds, expanding the catalog of known superintegrable systems.

Findings

Procedure effectively extends superintegrable systems on $ ext{E}^2$ and $ ext{S}^2$

Successfully applied to TTW and Calogero systems

Enhances understanding of superintegrability on constant curvature manifolds

Abstract

A procedure to extend a superintegrable system into a new superintegrable one is systematically tested for the known systems on $\mathbb E^2$ and $\mathbb S^2$ and for a family of systems defined on constant curvature manifolds. The procedure results effective in many cases including Tremblay-Turbiner-Winternitz and three-particle Calogero systems.