  Projective connections arise from equivalence classes of affine connections under the reparametrization of geodesics. They may also be viewed as quotient systems of the classical geodesic equation. After studying the link between integrals of the (classical) geodesic flow and its associated projective connection, we turn our attention to 2-dimensional metrics that admit one projective vector field, i.e. whose local flow sends unparametrized geodesics into unparametrized geodesics. We review and discuss the classification of these metrics, introducing special coordinates on the linear space of solutions to a certain system of partial differential equations, from which such metrics are obtained. Particularly, we discuss those that give rise to free second-order superintegrable Hamiltonian systems, i.e. which admit 2 additional, functionally independent quadratic integrals. We prove that these systems are parametrized by the 2-sphere, except for 6 exceptional points where the projective symmetry becomes homothetic. 
(Super-)integrable systems associated to 2-dimensional projective   connections with one projective symmetry