  We study the problem of constructing bulk and surface embedded modes (EMs)
inside the quasi-continuum band of a square lattice, using a potential
engineering approach \`a la Wigner and von Neumann. Building on previous
results for the one-dimensional (1D) lattice, and making use of separability,
we produce examples of two-dimensional envelope functions and the
two-dimensional (2D) potentials that produce them. The 2D embedded mode decays
like a stretched exponential, with a supporting potential that decays as a
power law. The separability process can cause that a 1D impurity state (outside
the 1D band) can give rise to a 2D embedded mode (inside the band). The
embedded mode survives the addition of random perturbations of the potential;
however, this process introduces other localized modes inside the band, and
causes a general tendency towards localization of the perturbed modes.
