Bi-Hamiltonian structure of the general heavenly equation

Abstract

We discover two additional Lax pairs and three nonlocal recursion operators for symmetries of the general heavenly equation introduced by Doubrov and Ferapontov. Converting the equation to a two-component form, we obtain Lagrangian and Hamiltonian structures of the two-component general heavenly system. We discover that in the two-component form we have only a single nonlocal recursion operator. Composing the recursion operator with the first Hamiltonian operator we obtain second Hamiltonian operator. Thus, the general heavenly equation in the two-component form is a bi-Hamiltonian system completely integrable in the sense of Magri.