Abstract
We present the derivation, for these vacuum metrics, of the Painlevé VI equation first obtained by Christodoulakis and Terzis, from the field equations for both minkowskian and euclidean signatures. This allows a complete discussion and the precise connection with some old results due to Kinnersley. The hyperkähler metrics are shown to belong to the Multi-Centre class and for the cases exhibiting an integrable geodesic flow the relevant Killing tensors are given. We conclude by the proof that for the Bianchi B family, excluding type III, there are no hyperkähler metrics.
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