At the first stage, the cooperative defect-deformational (DD) nucleation of seeding nanostructure occurs, described by the original deterministic DD Kuramoto-Sivashinsky (DDKS) equation for the concentration of mobile adatoms (surface defects). <...> Periodic surface relief, formed at the nucleation stage, related to the surface defect deformation potential, serves as a self-organized mask for subsequent growth of nanostructures, described by the convential deterministic Kardar-Parisi-Zhang (KPZ) equation for the relief height. <...> Computer simulations of nonlinear DDKS and KPZ equations describe the two-stage formation, in dependence on the sign of the surface defect deformation potential, of disordered and hexagonally ordered ensembles of nanoparticles or honeycomb void nanostructures. <...> The spatial DD harmonics interaction during cooperative nucleation is shown to play the key role in determination of the resulting nanostructures characteristics Index terms: atomic deposition, hexagonally ordered ensembles of nanoparticles and honeycomb void nanostructures formation, computer modeling 1. <...> A two-dimensional DDKS equation was derived for defect concentration in the defect-enriched surface solid layer, formed by laser irradiation [7], and its numerical solutions have been used for the interpretation of the formation of surface nano- and microstructures under the action of laser pulses [8]. <...> In the works [9-11] the conception of the DDKS equation was extended to the ensemble of interacting with each other adatoms. <...> First, in works [9,10], the theory of the surface defect-deformational (DD) instability in the system of mobile adatoms (surface defects), interacting via quasi-static surface acoustic (Rayleigh) waves has been developed. <...> An initial surface fluctuation strain gives rise to strain-induced surface defect fluxes. <...> This leads to the formation of the spatially nonuniform field of adatom concentration, which via the deformation potential of adatoms and also via the local renormalization of the surface energy, nonuniformly deforms the surface and the underlying elastic continuum, increasing thus the initial strain. <...> At exceeding of a certain critical value of the adatom concentration, such positive feedback leads to the onset of DD instability with arising of nanometer scale surface strain and relief modulations and piling of adatoms at extrema of the surface strain. <...> Then, in the work [11] it was shown that, under some restrictions, starting equations of the surface DD self-organization [9,10] can be reduced to a closed two-dimensional, nonlinear partial differential equation for adatom concentration of the KS equation type <...>