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Projective connections arise from equivalence classes of affine connectionsunder the reparametrization of geodesics. They may also be viewed as quotientsystems of the classical geodesic equation. After studying the link betweenintegrals of the (classical) geodesic flow and its associated projectiveconnection, we turn our attention to 2-dimensional metrics that admit oneprojective vector field, i.e. whose local flow sends unparametrized geodesicsinto unparametrized geodesics. We review and discuss the classification ofthese metrics, introducing special coordinates on the linear space of solutionsto a certain system of partial differential equations, from which such metricsare obtained. Particularly, we discuss those that give rise to freesecond-order superintegrable Hamiltonian systems, i.e. which admit 2additional, functionally independent quadratic integrals. We prove that thesesystems are parametrized by the 2-sphere, except for 6 exceptional points wherethe projective symmetry becomes homothetic.

(Super-)integrable systems associated to 2-dimensional projective connections with one projective symmetry