Symmetry Reduction of Asymmetric Heavenly Equation and 2+1-Dimensional Bi-Hamiltonian System

Abstract

The first heavenly equation of Plebanski in the two-component form is known to be a 3 + 1 role=presentation> 3 + 1 3 + 1 3+1 -dimensional tri-Hamiltonian system. We show that a particular choice of symmetry reduction applied to the first heavenly equation yields a 2 + 1 role=presentation> 2 + 1 2 + 1 2+1 -dimensional bi-Hamiltonian system. For this tri-dimensional system, we present Lagrangian, Hamiltonian, and recursion operators; point symmetries; and integrals of motions.