the previous results on the ring [@Popovych2011; @Yanchuk2011], extending them to the two-dimensional case. However, the analysis, which we have to employ has important differences. In particular, the combination of the spatial structure of the system (spatial coordinates $m$ and $n$) and temporal delays required the introduction of a so called “hybrid dispersion relation” for the investigation of the stability of stationary state and nonlinear plane waves in the homogeneous system. Roughly speaking, this hybrid dispersion relation is a synthesis of the dispersion relation from the pattern formation theory in spatially extended systems [@Cross1993; @Cross2009] and the pseudo-continuous spectrum developed for purely temporal delay systems [@Wolfrum2006; @Lichtner2011; @Wolfrum2010].

We believe that such a higher-dimensional extension allows to speculate about the possibility of employing such systems for generating or saving visual patterns, and can be probably of use for information processing purposes. Small arrays of delay-coupled optoelectronic oscillators have indeed already been realized experimentally [@Williams2013]. Similarly, autonomous Boolean networks of electronic logic gates have been demonstrated as versatile tools for the realizations of various space-time patterns [@Rosin2013]. Moreover, our analysis provides another evidence that the delays in coupled systems can play a constructive functional role.

More specifically, we consider a lattice 