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This paper combines two classical theories, namely metric projectivedifferential geometry and superintegrability. We study superintegrable systemson 2-dimensional geometries that share the same geodesics, viewed asunparametrized curves. We give a definition of projective equivalence of suchsystems, which may be considered the projective analog of (conformal) St\"ackelequivalence (coupling constant metamorphosis). Then, we discuss thetransformation behavior for projectively equivalent superintegrable systems andfind that the potential on a projectively equivalent geometry can bereconstructed from a characteristic vector field. Moreover, potentials ofprojectively equivalent Hamiltonians follow a linear superimposition rule. Thetechniques are applied to several examples. In particular, we use them toclassify, up to St\"ackel equivalence, the superintegrable systems ongeometries with one, non-trivial projective symmetry.

Projectively equivalent 2-dimensional superintegrable systems with projective symmetries