  We reconsider non-degenerate second order superintegrable systems in
dimension two as geometric structures on conformal surfaces. This extends a
formalism developed by the authors, initially introduced for
(pseudo-)Riemannian manifolds of dimension three and higher. The governing
equations of non-degenerate second order superintegrability in dimension two
are structurally significantly different from those valid in higher dimensions.
Specifically, we find conformally covariant structural equations, allowing one
to classify the (conformal classes of) non-degenerate second order
superintegrable systems on conformal surfaces geometrically. We then specialise
to second order properly superintegrable systems on surfaces with a
(pseudo-)Riemannian metric and obtain structural equations in accordance with
the known equations for Euclidean space. We finally give a single explicit set
of purely algebraic equations defining the variety parametrising such systems
on all constant curvature surfaces.
