  The CPT-even abelian gauge sector of the Standard Model Extension is
represented by the Maxwell term supplemented by
$(K_{F})_{\mu\nu\rho\sigma}F^{\mu\nu}F^{\rho\sigma}$, where the
Lorentz-violating background tensor, $(K_{F})_{\mu\nu\rho\sigma}$, possesses
the symmetries of the Riemann tensor. In the present work, we examine the
planar version of this theory, obtained by means of a typical dimensional
reduction procedure to $(1+2)$ dimensions. The resulting planar electrodynamics
is composed of a gauge sector containing six Lorentz-violating coefficients, a
scalar field endowed with a noncanonical kinetic term, and a coupling term that
links the scalar and gauge sectors. The dispersion relation is exactly
determined, revealing that the six parameters related to the pure
electromagnetic sector do not yield birefringence at any order. In this model,
the birefringence may appear only as a second order effect associated with the
coupling tensor linking the gauge and scalar sectors.The equations of motion
are written and solved in the stationary regime. The Lorentz-violating
parameters do not alter the asymptotic behavior of the fields but induce an
angular dependence not observed in the Maxwell planar theory.
