Generalization of bi-Hamiltonian systems in (3+1) dimension, possessing partner symmetries

We study bi-Hamiltonian structure of a general equation which possesses partner symmetries. The general form of such second-order PDEs with four independent variables was determined in the paper Sheftel and Malykh (2009) on a classification of second-order PDEs which have this property. We apply Dirac's theory of constraints to this general equation. We formulate the equation in a two-component form and present the Lax pair of Olver-Ibragimov-Shabat type. Under some constraints imposed on constant coefficients of this equation, we obtain its bi-Hamiltonian structure. Therefore, by Magri's theorem it is a completely integrable bi-Hamiltonian system in (3+1) dimensions. We also showed that with suitable choices of constant coefficients the equation is reduced to the well known integrable bi-Hamiltonian systems in (3+1) dimension.