Null Kähler Geometry and Isomonodromic Deformations

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