  In the present paper we study some new classical Lorentz violating effects in
planar electrodynamics due to the presence of stationary point-like field
sources. Starting from the Carroll-Field-Jackiw model defined in (3+1)
dimensions, which belongs to the electromagnetic CPT-odd sector of the Standard
Model Extension (SME), we apply the dimensional reduction procedure obtaining a
(2+1)-dimensional model that encompasses an electromagnetic sector composed of
the Maxwell-Chern-Simons electrodynamics, a pure scalar sector described by a
massless Klein-Gordon field, and a mixed sector where the background vector
mediates contributions involving both the scalar and the gauge fields. For all
the sectors of this planar theory, we explore some physical phenomena that
arise from the interactions between external sources. Specifically, we obtain
perturbative effects up to second order in the background vector related to the
presence of both electric and scalar planar charges and Dirac points.
