  We study the structure and stability of discrete breathers (both pinned and
mobile) in two-dimensional nonlinear anisotropic Schrodinger lattices. Starting
from a set of identical one-dimensional systems we develop the continuation of
the localized pulses from the weakly coupled regime (strongly anisotropic) to
the homogeneous one (isotropic). Mobile discrete breathers are seen to be a
superposition of a localized mobile core and an extended background of
two-dimensional nonlinear plane waves. This structure is in agreement with
previous results on onedimensional breather mobility. The study of the
stability of both pinned and mobile solutions is performed using standard
Floquet analysis. Regimes of quasi-collapse are found for both types of
solutions, while another kind of instability (responsible for the discrete
breather fission) is found for mobile solutions. The development of such
instabilities is studied, examining typical trajectories on the unstable
nonlinear manifold.
