  In this work, we explicitly evaluate the gauge propagator of the Maxwell
theory supplemented by the CPT-even term of the SME. First, we specialize our
evaluation for the parity-odd sector of the tensor $W_{\mu \nu \rho \sigma}$,\
using a parametrization that retains only the three nonbirefringent
coefficients. From the poles of the propagator, it is shown that physical modes
of this electrodynamics are stable, non-causal and unitary. In the sequel, we
carry out the parity-even gauge propagator using a parametrization that allows
to work with only the isotropic nonbirefringent element. In this case, we show
that the physical modes of the parity-even sector of the tensor $W$ are causal,
stable and unitary for a limited range of the isotropic coefficient.
