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 explicitly 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.