Einstein Metrics, Projective Structures and the $SU(infty)$ Toda Equation

We establish an explicit correspondence between two--dimensional projective structures admitting a projective vector field, and a class of solutions to the $SU(infty)$ Toda equation. We give several examples of new, explicit solutions of the Toda equation, and construct their mini--twistor spaces. Finally we discuss the projective-to-Einstein correspondence, which gives a neutral signature Einstein metric on a cotangent bundle $T^*N$ of any projective structure $(N, [ abla])$. We show that there is a canonical Einstein of metric on an $R^*$--bundle over $T^*N$, with a connection whose curvature is the pull--back of the natural symplectic structure from $T^*N$.