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 alpha-planes (algebraically special real alpha-geometry). In particular, we determine the behaviour of (four-dimensional neutral) Walker geometry under conformal rescalings and provide a derivation of the hyperheavenly equation from conformal rescaling formulae.

27 pages; v3 of 9 Sep 2011 replaces earlier versions, correcting minor typographical errors. The published version is slightly abbreviated