  Superintegrable systems are classical and quantum Hamiltonian systems which
enjoy much symmetry and structure that permit their solubility via analytic and
even, algebraic means. They include such well-known and important models as the
Kepler potential, Calogero-Moser model, and harmonic oscillator, as well as its
integrable perturbations, for example, the Smorodinsky-Winternitz (SW)
potential. Normally, the problem of classification of superintegrable systems
is approached by considering associated algebraic, or geometric structures. To
this end we invoke the invariant theory of Killing tensors (ITKT). Through the
ITKT, and in particular, the recursive version of the Cartan method of moving
frames to derive joint invariants, we are able to intrinsically characterise
and interpret the arbitrary parameters appearing in the general form of the SW
superintegrable potential. Specifically, we determine, using joint invariants,
that the more general the geometric structure associated with the SW potential
is, the fewer arbitrary parameters it admits. Additionally, we classify the
multi-separability of a recently discovered superintegrable system which
generalizes the SW potential and is dependent on an additional arbitrary
parameter k, known as the Tremblay-Turbiner-Winternitz (TTW) system. We provide
a proof that only for the case k=+/-1 does the general TTW system admit
orthogonal separation of variables with respect to both Cartesian and polar
coordinates.
