  We construct the normal forms of null-K\"ahler metrics: pseudo-Riemannian
metrics admitting a compatible parallel nilpotent endomorphism of the tangent
bundle. Such metrics are examples of non-Riemannian holonomy reduction, and (in
the complexified setting) appear in the Bridgeland stability conditions of the
moduli spaces of Calabi-Yau three-folds.
  Using twistor methods we show that, in dimension four - where there is a
connection with dispersionless integrability - the cohomogeneity-one
anti-self-dual null-K\"ahler metrics are generically characterised by solutions
to Painlev\'e I or Painlev\'e II ODEs.
