A Kähler Compatible Moyal Deformation of the First Heavenly Equation

We construct a noncommutative Kähler manifold based on a non-linear perturbations of Moyal integrable deformations of $D=4$ self-dual gravity. The deformed Kähler manifold preserves all the properties of the commutative one, and we obtain the associated noncommutative Kähler potential using the Moyal deformed gravity approach. We apply this construction to the Atiyah–Hitchin metric and its Kähler potential, which is useful in the description of interactions among magnetic monopoles at low energies.