Nonlinear dynamics on the plane and integrable hierarchies of infinitesimal deformations

A class of nonlinear problems on the plane, described by nonlinear inhomogeneous ∂¯-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 ∂¯-dressing method