  We examine one- and two-dimensional (1D and 2D) models of linearly coupled
lattices of the discrete-nonlinear-Schr{\"{o}}dinger type. Analyzing ground
states of the systems with equal powers in the two components, we find a
symmetry-breaking phenomenon beyond a critical value of the squared $l^2$-norm.
Asymmetric states, with unequal powers in their components, emerge through a
subcritical pitchfork bifurcation, which, for very weakly coupled lattices,
changes into a supercritical one. We identify the stability of various solution
branches. Dynamical manifestations of the symmetry breaking are studied by
simulating the evolution of the unstable branches. The results present the
first example of spontaneous symmetry breaking in 2D lattice solitons. This
feature has no counterpart in the continuum limit, because of the collapse
instability in the latter case.
