Superintegrable extensions of superintegrable systems

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.