A Note on Weak ${\cal H}$-Spaces, Heavenly Equations and their Evolution Weak Heavenly Equations

Abstract

\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.