Symmetries, integrals, and three-dimensional reductions of Plebañski’s second heavenly equation

Abstract

We study symmetries and conservation laws for Plebañski’s second heavenly equation written as a first-order nonlinear evolutionary system which admits a multi-Hamiltonian structure. We construct an optimal system of one-dimensional subalgebras and all inequivalent three-dimensional symmetry reductions of the original four-dimensional system. We consider these two-component evolutionary systems in three dimensions as natural candidates for integrable systems.