On the ∂̄-dressing method applicable to heavenly equation

The ∂̄-dressing scheme based on local nonlinear vector ∂̄-problem is developed. It is applicable to multidimensional nonlinear equations for vector fields, and, after Hamiltonian reduction, to heavenly equation. Hamiltonian reduction is described explicitely in terms of the ∂̄-data. An analogue of Hirota bilinear identity for heavenly equation hierarchy is introduced, τ -function for the hierarchy is defined. Addition formulae (generating equations) for the τ -function are found. It is demonstrated that τ -function for heavenly equation hierarchy is given by the action for ∂̄-problem evaluated on the solution of this problem.