  The Lax formulation of the hyper-Hermiticity condition in four dimensions is
used to derive a potential that generalises Plebanski's second heavenly
equation for hyper-Kahler 4-manifolds. A class of examples of hyper-Hermitian
metrics which depend on two arbitrary functions of two complex variables is
given. The twistor theory of four-dimensional hyper-Hermitian manifolds is
formulated as a combination of the Nonlinear Graviton Construction with the
Ward transform for anti-self-dual Maxwell fields.
