Symmetry reduction of asymmetric heavenly equation and bi-Hamilton structure

In this work we study 3+1-dimensional bi-Hamiltonian integrable systems which are developed considerably in recent years. In general we are interested in nonlinear second order differential equations with one dependent variable and four independent variables x, y, z and t. One of the examples of these equations is asymmetric heavenly equation, obtained as one of the canonical equations in the classification of nonlinear second order partial diferantial equations that possess partner symmetries. We show that one symmetry reduction of this equation yields a new 2+1-dimensional multi-Hamiltonian intagrable system. It is shown that reduced 2+1-dimensional system set in a two-component form admits a bi-Hamiltonian structure. For a two-component system a new Lagrangian is introduced and Dirac's constraint theory is applied to obtain the symplectic and first Hamiltonian structure. In order to get Frechet derivative of the system the compatibility condition is applied to the Lie group transformation which is only considered for a dependent variable. It is also shown that the commutator of the recursion operator and Frechet derivative reproduce the system and therefore form a Lax pair Olver-Ibragimov-Shabat type. Then the Recursion operator is constructed and the second Hamiltonian structure for this system is obtained by applying the recursion operator on the first Hamiltonian operator. Finally Jacobi identity for Hamiltonian structure is proven by using P. Olver's method. Therefore we conclude that reduced new 2+1-dimensional system is completely integrable by Margi's theorem.