  We pursue the investigation of a generic non-linear extension of axionic
electrodynamics in a Carroll-Field-Jackiw (CFJ) scenario that implements
Lorentz-symmetry violation (LSV). The model we inspect consists of an arbitrary
non-linear electrodynamic action coupled to the axion field in presence of an
anisotropy four-vector that realizes the breaking of Lorentz symmetry under the
particle point of view. The non-linear electromagnetic field is expanded around
a constant and uniform magnetic background up to second order in the
propagating photon field. The focus of our attention is the study of the
material properties of the vacuum in the particular case of a space-like CFJ
$4$-vector. The dispersion relations associated to the plane wave solutions are
explicitly worked out in two situations: the magnetic background perpendicular
and parallel to the wave direction. We extend these results to consider the
analysis of the birefringence phenomenon in presence of non-linearity, the
axion and the LSV manifested through the spatial anisotropy. Three specific
proposals of non-linear electrodynamics are contemplated: Euler-Heisenberg,
Born-Infeld and the Modified Maxwell electrodynamics. Throughout the paper, we
shall justify why we follow the unusual path of connecting, in a single
Lagrangian density, three pieces of physics beyond the Standard Model, namely,
non-linearity, axions and LSV. Our true goal is to actually inspect and
describe how axionic, non-linear and LSV effects interfere with one another
whenever physical entities like group velocity, refraction indices,
birefringence and effective masses of physical excitations are computed in
presence of an external constant and homogeneous magnetic field.
