Abstract
We provide an affirmative answer to a question posed by Tod (1995, Twistor Theory (New York: Dekker)), and construct all four-dimensional Kähler metrics with vanishing scalar curvature which are invariant under the conformal action of the Bianchi V group. The construction is based on the combination of twistor theory and the isomonodromic problem with two double poles. The resulting metrics are non-diagonal in the left-invariant basis and are explicitly given in terms of Bessel functions and their integrals. We also make a connection with the LeBrun ansatz, and characterize the associated solutions of the SU(∞) Toda equation by the existence a non-abelian two-dimensional group of point symmetries.
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