  In this short note we discuss a new formula for solving the nonlocal
$\overline{\partial}$-problem, and discuss application to the Manakov--Zakharov
dressing method. We then explicitly apply this formula to solving the complex
(2+1)D Kadomtsev--Petviashvili equation and complex (2+1)D completely
integrable generalization of the (2+1)D Kaup--Broer (or Kaup--Boussinesq)
system. We will also discuss how real (1+1)D solutions are expressed using this
formalism. It is simple to express the formalism for finite gap primitive
solutions from [10], [8] using the formalism of this note. We also discuss
recent results on the infinite soliton limit for the (1+1)D Korteweg--de Vries
equation and the (2+1)D Kaup--Broer system. In an appendix, the classical
solutions to the 3D Laplace equation (2+1)D d'Alembert wave equation by
Whittaker are described. This appendix is included to elucidate an analogy
between the dressing method and the Whittaker solutions.
