A necessary and sufficient condition for a second-order superintegrable system with $(n+1)$-parameter potential to extend to a non-degenerate second-order superintegrable system

The isotropic harmonic oscillator and the Kepler-Coulomb system are two examples of second-order (maximally) superintegrable (Hamiltonian) systems, and they are pivotal models in the Sciences. They are classified in dimension two and, partially, in dimension three. We obtain a necessary and sufficient condition for second-order superintegrable systems with a $(n+1)$-parameter potential to extend to so-called non-degenerate systems, i.e.\ systems with a larger family of compatible potentials. Concretely, we show that a $(n+1)$-parameter potential is the restriction of a non-degenerate potential if and only if a certain trace-free tensor field vanishes.
Comment: 16 pages