  Taking as starting point the planar model arising from the dimensional
reduction of the Abelian-Higgs Carroll-Field-Jackiw model, we write down and
study the extended Maxwell equations and the corresponding wave equations for
the potentials. The solutions for these equations correspond to the usual ones
for the MCS-Proca system, supplemented with background-dependent correction
terms. In the case of a purely timelike background, exact algebraic solutions
are presented which possess a similar behavior to the MCS-Proca counterparts
near and far from the origin. On the other hand, for a purely spacelike
background, only approximate solutions are feasible. They consist of
non-trivial analytic expressions with a manifest evidence of spatial
anisotropy, which is consistent with the existence of a privileged direction in
space. These solutions also behave similarly to the MCS-Proca ones near and far
from the origin.
