  Based on the motivation that some quantum gravity theories predicts the
Lorentz Invariance Violation (LIV) around Planck-scale energy levels, this
paper proposes a new formalism that addresses the possible effects of LIV in
the electrodynamics. This formalism is capable of changing the usual
electrodynamics through high derivative arbitrary mass dimension terms that
includes a constant background field controlling the intensity of LIV in the
models, producing modifications in the dispersion relations in a manner that is
similar to the Myers-Pospelov approach. With this framework, it was possible to
generate a CPT-even and CPT-odd generalized modifications of the
electrodynamics in order to study the stability and causality of these theories
considering the isotropic case for the background field. An additional analysis
of unitarity at tree level was considered by studying the saturated
propagators. After this analysis, we conclude that, while CPT-even
modifications always preserves the stability, causality and unitarity in the
boundaries of the effective field theory and therefore may be good candidates
for field theories with interactions, the CPT-odd one violates causality and
unitarity. This feature is a consequence of the vacuum birefringence
characteristics that are present in CPT-odd theories for the photon sector.
