Fine structure for 3D second-order superintegrable systems: Three-parameter potentials

A classical (or quantum) superintegrable system of second order is an integrable n-dimensional Hamiltonian system with potential that admits 2n - 1 functionally independent constants of the motion quadratic in the momenta, the maximum possible. For n ≤ 3 on conformally flat spaces with nondegenerate, i.e., four-parameter potentials (the extreme case), we have worked out the structure and classified most of the possible spaces and potentials. Here, we extend the analysis to a more degenerate class of functionally linearly independent superintegrable systems, the three-parameter potential case. We show that for 'true' three-parameter potentials the algebra of constants of the motion no longer closes at order 6 but still all such systems are Stäckel transforms of systems on complex Euclidean space or the complex 3-sphere. This is a significant step towards the complete structure analysis of all types of second-order superintegrable systems.