  We demonstrate how the complex integral formula for the Airy functions arises
from Penrose's twistor contour integral formula. We then use the Lax
formulation of the isomonodromy problem with one irregular singularity of order
four to show that the Airy equation arises from the anti-self-duality equations
for conformal structures of neutral signature invariant under the isometric
action of the Bianchi II group. This conformal structure admits a null-K\"ahler
metric in its conformal class which we construct explicitly.
