  On spaces of constant curvature, the geodesic equations automatically have
higher order integrals, which are just built out of first order integrals,
corresponding to the abundance of Killing vectors. This is no longer true for
general conformally flat spaces, but in this case there is a large algebra of
conformal symmetries. In this paper we use these conformal symmetries to build
higher order integrals for the geodesic equations. We use this approach to give
a new derivation of the Darboux-Koenigs metrics, which have only one Killing
vector, but two quadratic integrals. We also consider the case of possessing
one Killing vector and two cubic integrals.
  The approach allows the quantum analogue to be constructed in a simpler
manner.
