Following Plebański and Robinson, complex V 4 ’s which admit a congruence of totally null surfaces are shown to have coordinates which, in pairs, have a spinor structure which generates the usual spinor structure of the 2‐forms over the space. This structure allows Einstein’s vacuum equations to fracture into three triples and a singlet, which allow for easy reduction of the entire set to one nonlinear partial differential equation needed for consistency. An inhomogeneous GL (2, C ) group of coordinate transformations, constrained to leave the tetrad form invariant, is constructed and used to simplify the equations and clarify the geometrical meaning of the parameters introduced during the integration process.
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