New heavenly double copies

The double copy relates scattering amplitudes and classical solutions in Yang-Mills theory, gravity, related field theories. Previous work has shown that this an explicit realisation self-dual YM where the equation of motion can be written a form maps directly to Pleba\'nski's heavenly for gravity. involves area-preserving diffeomorphism algebra, two copies which appear equation. In paper, we show construction is special case wider family heavenly-type examples, by (i) performing Moyal deformations, (ii) replacing diffeomorphisms with less restricted algebra. As result, obtain double-copy interpretation hyper-Hermitian manifolds, extending previously known hyper-K\"ahler case. We also introduce double-Moyal deformation examples Lax pairs possible are manifestly consistent Ward's conjecture, suggest integrability gravity-type theory may guaranteed general at least one gauge-theory-type single copies.