Symmetry Reductions of Second Heavenly Equation and 2+1-Dimensional Hamiltonian Integrable Systems

Abstract

Second heavenly equation of Plebañski, presented in a two-component form, is known to be a 3 + 1-dimensional multi-Hamiltonian integrable system. We show that one symmetry reduction of this equation yields a two component 2 + 1-dimensionalmulti-Hamiltonian integrable system. For this system, we present Hamiltonian and recursion operators, point symmetries and integrals of motion. For another symmetry reduction, the reduced system is ”almost bi-Hamiltonian”, with two known Hamiltonian operators but the second Hamiltonian density missing.