%0 Journal Article
%T Further results on non-diagonal Bianchi type III vacuum metrics
%A Galliano Valent
%J Physics
%D 2011
%I arXiv
%R 10.1007/s10714-012-1340-9
%X We present the derivation, for these vacuum metrics, of the Painlev\'e 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 hyperk\"ahler 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 hyperk\"ahler metrics.
%U http://arxiv.org/abs/1102.4538v1