Second-order superintegrable systems and Weylian geometry

Abundant second-order maximally conformally superintegrable Hamiltonian systems are re-examined, revealing their underlying natural Weyl structure and offering a clearer geometric context for the study of St\"ackel transformations (also known as coupling constant metamorphosis). This also allows us to naturally extend the concept of conformal superintegrability from the realm of conformal geometries to that of Weyl structures. It enables us to interpret superintegrable systems of the above type as semi-Weyl structures, a concept related to statistical manifolds and affine hypersurface theory.