  Quaternion-Kaehler four-manifolds, or equivalently anti-self-dual Einstein
manifolds, are locally determined by one scalar function subject to
Przanowski's equation. Using twistorial methods we construct a Lax Pair for
Przanowski's equation, confirming its integrability. The Lee form of a
compatible local complex structure, which one can always find, gives rise to a
conformally invariant differential operator acting on sections of a line
bundle. Special cases of the associated generalised Laplace operator are the
conformal Laplacian and the linearised Przanowski operator. We provide
recursion relations that allow us to construct cohomology classes on twistor
space from solutions of the generalised Laplace equation. Conversely, we can
extract such solutions from twistor cohomology, leading to a contour integral
formula for perturbations of Przanowski's function. Finally, we illuminate the
relationship between Przanowski's function and the twistor description, in
particular we construct an algorithm to retrieve Przanowski's function from
twistor data in the double-fibration picture. Using a number of examples, we
demonstrate this procedure explicitly.
