  In this work, we study the radiative generation of the Lorentz-violating
Euler-Heisenberg action, in the weak field approximation. For this, we first
consider a nonperturbative calculation in the coefficient $c_{\mu\nu}$,
however, by assuming rotational invariance. Within this approach, we also
obtain the results of the amplitude for the photon triple splitting. In the
following, we take into account the perturbative approach, where $c_{\mu\nu}$
is treated as a insertion in the propagator and a new vertex. The partial
results are in fact an expansion up to first order in $c_{\mu\nu}$ of the
nonperturbative ones, with $c_{00}=\kappa$ and $c_{0i}=c_{ij}=0$. This suggest
that the complete results obtained in the nonperturbative approach can be used
in both treatments.
