  The classical Kepler-Coulomb system in 3 dimensions is well known to be 2nd
order superintegrable, with a symmetry algebra that closes polynomially under
Poisson brackets. This polynomial closure is 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 extended Kepler-Coulomb system (also 2nd order
superintegrable) is an exception, as its quadratic symmetry algebra doesn't
close polynomially. The 3D 4-parameter potential for the extended
Kepler-Coulomb system is not even 2nd order superintegrable. However, Verrier
and Evans (2008) showed it was 4th order superintegrable, and Tanoudis and
Daskaloyannis (2011) showed that in the quantum case, if a second 4th order
symmetry is added to the generators, the double commutators in the symmetry
algebra close polynomially. Here, based on the Tremblay, Turbiner and
Winternitz construction, we consider an infinite class of classical 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 work out explicitly
the structure of the symmetry algebras determined by the 5 basis generators we
have constructed. We demonstrate that the symmetry algebras close rationally;
only for systems admitting extra discrete symmetries is polynomial closure
achieved. Underlying the structure theory is the existence of raising and
lowering constants of the motion, not themselves polynomials in the momenta,
that can be employed to construct the polynomial symmetries and their structure
relations.
