Canonical reductions of the TED equation: integrable deformations of heavenly-type equations

Natural classes of integrability-preserving reductions of a 4+4-dimensional generalisation (TED equation) of the general heavenly equation are recorded. In particular, these reductions lead to integrable ‘deformations’ of various other avatars of the heavenly equation governing self-dual Einstein spaces. The known deformed heavenly equations which give rise to half-flat conformal structures are retrieved in this manner. Moreover, Plebański’s link between the first and second heavenly equations is extended to their deformed counterparts.