Twistor construction of asymptotically hyperbolic Einstein-Weyl spaces

Starting from a real analytic conformal Cartan connection on a real analytic surface S, we construct a complex surface T containing a family of pairs of projective lines. Using the structure on S we also construct a complex 3-space Z, such that Z is a twistor space of a self-dual conformal 4-fold and T is a quotient of Z by a holomorphic local C× action. We prove that T is a minitwistor space of an asymptotically hyperbolic Einstein-Weyl space with S as an asymptotic boundary.