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Self-dual Bianchi metrics and the Painlevé transcendents

We give two constructions that relate the self-duality condition on the Weyl curvature of a spacetime with Bianchi IX symmetry to the Painlevé equations, the first by way of the self-dual Yang--Mills equation, the second by way of the isomonodromy problem. We explain how the generic self-dual conformal structure with this symmetry can be obtained from the general solution of the sixth Painlevé equation.