  We have studied a (1+2)-dimensional Lorentz-violating model which is obtained
from the dimensional reduction of the nonbirefringent sector of the CPT-even
electrodynamics of the standard model extension (SME). The planar theory
contains a gauge sector and a scalar sector which are linearly coupled by means
of a Lorentz-invariance violating (LIV) vector, $S^{\mu}$, while the kinetic
terms of both sectors are affected by the components of a Lorentz-violating
symmetric tensor, $\kappa^{\mu\nu}$. The energy-momentum tensor reveals that
both sectors present energy stability for sufficiently small values of the
Lorentz-violating parameters. The full dispersion relation equations are
exactly determined and analyzed for some special configurations of the LIV
backgrounds, showing that the planar model is entirely nonbirefringent at any
order in the LIV parameters. At first order, the gauge and scalar sectors are
described by the same dispersion relations. Finally, the equations of motion
have been solved in the stationary regime and at first order in the LIV
parameters. It is observed that the Lorentz-violating parameters do not alter
the asymptotical behavior of the electric and magnetic fields but induce an
angular dependence which is not present in Maxwell's planar theory.
