Doubrov-Ferapontov general heavenly equation and the hyper-K\"ahler hierarchy

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.