Algebraically special, real alpha-geometries

We exploit the spinor description of four-dimensional Walker geometry, and conformal rescalings of such, to describe the local geometry of four-dimensional neutral geometries with algebraically degenerate self-dual Weyl curvature and an integrable distribution of αα-planes (algebraically special real  αα-geometry). In particular, we determine the behaviour of Walker geometry under conformal rescaling and provide a derivation of the hyperheavenly equation from conformal rescaling formulae.

► For four-dimensional neutral geometry ► algebraically degenerate SD Weyl curvature with integrable distribution of αα-planes ► characterized locally as conformally Walker ► spinor methods and conformal rescaling formulae yield hyperheavenly equation.