Mixed heavenly equation and Husain’s equation are integrable bi-Hamiltonian systems

In the recent paper by one of the authors (MBS) and A. A. Malykh on the classification of second-order PDEs with four independent variables that possess partner symmetries [1], mixed heavenly equation and Husain’s equation appear as closely related canonical equations admitting partner symmetries. Here for the mixed heavenly equation and Husain’s equation, formulated in a two-component form, we present Lax pairs of Olver-Ibragimov-Shabat type and obtain their multi-Hamiltonian structure. Thus, we discover new bi-Hamiltonian systems, integrable in the sense of Magri.