Bi-Hamiltonian structure of an asymmetric heavenly equation

Abstract

In the paper of Sheftel and Malykh (2009 J. Phys. A: Math. Theor. 42 395202) on the classification of second-order PDEs with four independent variables that possess partner symmetries, an asymmetric heavenly equation appears as one of the canonical equations admitting partner symmetries. Here, for the asymmetric heavenly equation formulated in a two-component form, we present the Lax pair of Olver-Ibragimov-Shabat type and obtain its multi-Hamiltonian structure. Therefore, by Magri's theorem, it is a completely integrable bi-Hamiltonian system in four dimensions.