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.Comment: 23 pages, 5 table
