  We give a description of recently introduced Doubrov-Ferapontov general
heavenly equation in terms of closed differential Pl\"ucker two-form,
rationally depending on the spectral parameter. We demonstrate that general
heavenly equation is an important generating equation in the context of
Takasaki hyper-K\"ahler hierarchy, and it is also directly connected to
hyper-K\"ahler geometry through the Gindikin construction. We develop a
$\bar{\partial}$-dressing scheme and introduce a formula for the potential
satisfying the general heavenly equation. Multidimensional generalization is
also outlined.
