Anti-self-dual Hermitian metrics and Painlevé III (Q1763038)

Cohomogeneity-one metrics with an \(\text{SU}(2)\) isometry group transitive on codimension one surfaces with self-dual Weyl tensor have been studied by many authors. It was shown that the self-dual Einstein equations for such metrics reduce to a particular Painlevé VI equation. Painlevé III equations are degenerated from Painlevé VI. In this paper, the author studies the \(\text{SU}(2)\)-invariant anti-self-dual metrics which are specified by the solutions of Painlevé III. Diagonal and non-diagonal metrics are studied. The author shows that the metric is specified by a solution of Painlevé III if and only if there exists an \(\text{SU}(2)\)-invariant Hermitian structure.