  We take as starting point the planar model arising from the dimensional
reduction of the Maxwell Electrodynamics with the (Lorentz-violating)
Carroll-Field-Jackiw term. We then write and study the extended Maxwell
equations and the corresponding wave equations for the potentials. The solution
to these equations show some interesting deviations from the usual MCS
Electrodynamics, with background-dependent correction terms. In the case of a
time-like background, the correction terms dominate over the MCS sector in the
region far from the origin, and establish the behaviour of a massless
Electrodynamics (in the electric sector). In the space-like case, the solutions
indicate the clear manifestation of spatial anisotropy, which is consistent
with the existence of a privileged direction is space.
