Abstract
Motivated by the Bianchi-type-IX scalar-flat Kähler metric of Pedersen and Poon, an ansatz is presented which reduces the -Toda field equation to special cases of the Painlevé-III differential equation. This leads to new four-dimensional, scalar-flat, Kähler and hyper-Kähler metrics based on Painlevé transcendents. By considering the relation between Einstein--Weyl spaces and the -Toda field equation, we classify separable solutions of the latter equation and then characterize those Einstein--Weyl spaces which arise from it.
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