  A class of nonlinear problems on the plane, described by nonlinear
inhomogeneous $\bar{\partial}$-equations, is considered. It is shown that the
corresponding dynamics, generated by deformations of inhomogeneous terms
(sources) is described by Hamilton-Jacobi type equations associated with
hierarchies of dispersionless integrable systems. These hierarchies are
constructed by applying the quasiclassical $\bar{\partial}$-dressing method.
