Self-duality and the Painleve transcendents

It is shown that the six Painleve transcendental equations are essentially equivalent to the SL(2,C) self-dual Yang-Mills equations with certain three-dimensional Abelian groups of conformal symmetries. In the case of P VI , the equivalence arises from a direct twistor transform of the corresponding isomonodromy problem. A similar transform arises in the other cases, but indirectly by way of the deformation equations.