  We prove there is a unique vacuum solution in split-signature spacetimes with
Kleinian SO(2,1) spherical symmetry. We extend our analysis to accommodate a
positive or negative cosmological constant and we prove the Kleinian
spherically symmetric solutions to Einstein's equation are locally isomorphic
to the split-signature analogues of Schwarzschild-(Anti)-de Sitter or Nariai
spacetimes. Our analysis provides a Kleinian extension of Birkhoff's theorem to
metrics with split-signature. Axisymmetric vacuum solutions are also
considered, including (2,2) signature formulations of the Kerr and Taub-NUT
metrics.
