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In the recent paper by one of the authors (MBS) and A. A. Malykh on theclassification of second-order PDEs with four independent variables thatpossess partner symmetries (J. Phys. A: Math. Theor. Vol. 42 (2009) 395202(20pp)), mixed heavenly equation and Husain equation appear as closely relatedcanonical equations admitting partner symmetries. Here for the mixed heavenlyequation and Husain equation, formulated in a two-component form, we presentrecursion operators, Lax pairs of Olver-Ibragimov-Shabat type and discovertheir Lagrangians, symplectic and bi-Hamiltonian structure. We obtain all pointand second-order symmetries, integrals and bi-Hamiltonian representations ofthese systems and their symmetry flows together with infinite hierarchies ofnonlocal higher symmetries.

Bi-Hamiltonian representation, symmetries and integrals of mixed heavenly and Husain systems