  Using a combination of techniques from conformal and complex geometry, we
show the potentialization of 4-dimensional closed Einstein-Weyl structures
which are half-algebraically special and admit a "half-integrable"
almost-complex structure. That is, we reduce the Einstein-Weyl equations to a
single, conformally invariant, non-linear scalar equation, that we call the
"conformal HH equation", and we reconstruct the conformal structure (curvature
and metric) from a solution to this equation. We show that the conformal metric
is composed of: a conformally flat part, a conformally half-flat part related
to certain "constants" of integration, and a potential part that encodes the
full non-linear curvature, and that coincides in form with the Hertz potential
from perturbation theory. We also study the potentialization of the Dirac-Weyl,
Maxwell (with and without sources), and Yang-Mills systems. We show how to deal
with the ordinary Einstein equations by using a simple trick. Our results give
a conformally invariant, coordinate-free, generalization of the hyper-heavenly
construction of Plebanski and collaborators.
