  Second-order (maximally) conformally superintegrable systems play an
important role as models of mechanical systems, including systems such as the
Kepler-Coulomb system and the isotropic harmonic oscillator. The present paper
is dedicated to understanding non- and semi-degenerate systems. We obtain
"projective flatness" results for two torsion-free connections naturally
associated to such systems. This viewpoint sheds some light onto the
interrelationship of properly and conformally (second-order maximally)
superintegrable systems from a geometrical perspective. It is shown that the
semi-degenerate secondary structure tensor can be viewed as the Ricci curvature
of a natural torsion-free connection defined by the primary structure tensor
(and similarly in the non-degenerate case). It is also shown that properly
semi-degenerate systems are characterised, similar to the non-degenerate case,
by the vanishing of the secondary structure tensor.
