  A non-degenerate second-order maximally conformally superintegrable system in
dimension 2 naturally gives rise to a quadric with position dependent
coefficients. It is shown how the system's St\"ackel class can be obtained from
this associated quadric.The St\"ackel class of a second-order maximally
conformally superintegrable system is its equivalence class under St\"ackel
transformations, i.e., under coupling-constant metamorphosis.
