We represent a version of the dressing method allowing one to construct
a new class of n-wave type nonlinear partial differential equations (PDEs) whose solution space may be parametrized by arbitrary functions of several variables. The algorithm is based on the integral equation of special type. Its reduction to the classical D-bar problem is discussed. There is no restrictions on the dimensionality of nonlinear PDEs. The associated solution manifold is parametrized by the arbitrary functions of several variables, but this freedom is not enough to provide the full
integrability of nonlinear PDE. Some integrable reductions of the derived nonlinear
system are studied. We also represent matrix multi-lump 5-dimensional particular solutions to the reduced system.