  In this paper we continue the work of Kalnins et al in classifying all
second-order conformally-superintegrable (Laplace-type) systems over
conformally flat spaces, using tools from algebraic geometry and classical
invariant theory. The results obtained show, through Staeckel equivalence, that
the list of known nondegenerate superintegrable systems over three-dimensional
conformally flat spaces is complete. In particular, a 7-dimensional manifold is
determined such that each point corresponds to a conformal class of
superintegrable systems. This manifold is foliated by the nonlinear action of
the conformal group in three-dimensions. Two systems lie in the same conformal
class if and only if they lie in the same leaf of the foliation. This foliation
is explicitly described using algebraic varieties formed from representations
of the conformal group. The proof of these results rely heavily on Groebner
basis calculations using the computer algebra software packages Maple and
Singular.
