  The quantum Kepler-Coulomb system in 3 dimensions is well known to be 2nd
order superintegrable, with a symmetry algebra that closes polynomially under
commutators. This polynomial closure is also typical for 2nd order
superintegrable systems in 2D and for 2nd order systems in 3D with
nondegenerate (4-parameter) potentials. However the degenerate 3-parameter
potential for the 3D Kepler-Coulomb system (also 2nd order superintegrable) is
an exception, as its symmetry algebra doesn't close polynomially. The 3D
4-parameter potential for the extended Kepler-Coulomb system is not even 2nd
order superintegrable, but Verrier and Evans (2008) showed it was 4th order
superintegrable, and Tanoudis and Daskaloyannis (2011) showed that, if a 2nd
4th order symmetry is added to the generators, the symmetry algebra closes
polynomially. Here, based on the Tremblay, Turbiner and Winternitz
construction, we consider an infinite class of quantum extended Kepler-Coulomb
3 and 4-parameter systems indexed by a pair of rational numbers (k_1,k_2) and
reducing to the usual systems when k_1=k_2=1. We show these systems to be
superintegrable of arbitrarily high order and determine the structure of their
symmetry algebras. We demonstrate that the symmetry algebras close
algebraically; only for systems admitting extra discrete symmetries is
polynomial closure achieved. Underlying the structure theory is the existence
of raising and lowering operators, not themselves symmetry operators or even
defined independent of basis, that can be employed to construct the symmetry
operators and their structure relations.
