  We show that the definition of a second order superintegrable system on a
(pseudo-)Riemannian manifold gives rise to a conformally invariant notion of
superintegrability. Conformal equivalence is the natural extension of the
well-known St\"ackel transform, which in turn originates from the classical
Maupertuis-Jacobi principle. We extend our recently developed algebraic
geometric approach for the classification of second order superintegrable
systems in arbitrarily high dimension to conformally superintegrable systems,
which are presented via conformal scale choices of second order superintegrable
systems defined within a conformal geometry.
  For superintegrable systems on constant curvature spaces, we find that the
conformal scales of St\"ackel equivalent systems arise from eigenfunctions of
the Laplacian and that their equivalence is characterised by a conformal
density of weight two.
  Our approach yields an algebraic equation that governs the classification
under conformal equivalence for a prolific class of second order conformally
superintegrable systems. This class contains all non-degenerate examples known
to date, and is given by a simple algebraic constraint of degree two on a
general harmonic cubic form. In this way the yet unsolved classification
problem is put into the reach of algebraic geometry and geometric invariant
theory. In particular, no obstruction exists in dimension three, and thus the
known classification of conformally superintegrable systems is reobtained in
the guise of an unrestricted univariate sextic. In higher dimensions, the
obstruction is new and has never been revealed by traditional approaches.
