  \noindent It is find a non-linear partial differential equation which we show
contains the first Heavenly equation of Self-dual gravity and generalize the
second one. This differential equation we call "Weak Heavenly Equation" (${\cal
WH}$-equation). For the two-dimensional case the ${\cal WH}$-equation is
brought into the evolution form (Cauchy-Kovalevski form) using the Legendre
transformation. Finally, we find that this transformed equation ("Evolution
Weak Heavenly Equation") does admit very simple solutions.
