  We show that on a surface locally every affine torsion-free connection is
projectively equivalent to a Weyl connection. First, this is done using
exterior differential system theory. Second, this is done by showing that the
solutions of the relevant PDE are in one-to-one correspondence with the
sections of the `twistor' bundle of conformal inner products having holomorphic
image. The second solution allows to use standard results in algebraic geometry
to show that the Weyl connections on the two-sphere whose geodesics are the
great circles are in one-to-one correspondence with the smooth quadrics without
real points in the complex projective plane.
