  We investigate physical consequences of non-linear electrodynamic coupled to
parameters that signal violation Lorentz-symmetry breaking (LSV). Our
undertaking is done by considering a general non-linear photonic Lagrangian
which coupled to the Carroll-Field-Jackiw's model (CFJ). Our endeavor reveals
how the (meta) material constitutive properties of the vacuum and wave
propagation are affected by the interference of the LSV parameters LSV with the
specific non-linear electrodynamic model under consideration. We also discuss
the refractive indices for this new medium characterized by the coupling
between non-linearities and the operators that carry the LSV message. Our
results show that the QED-vacuum responds with birefringence and a dispersive
propagation of waves. Subsequently, we consider the electromagnetic radiation
produced by a moving charged particle interacting with this new medium. Our
inspection illustrates that the emitted radiation reproduces the features of
the Cherenkov effect for certain intensities of background magnetic fields .
Finally, we compute the static potential profile within the framework of the
gauge-invariant, but path-dependent, variables formalism. A logarithmic
correction to the usual static Coulomb potential emerges driven by the LSV
parameter and there also appear corrections due to the non-linearity;
nevertheless, the logarithm behavior drops out whenever the LSV parameter is
switched off.
