  We examine the nonperturbative structure of the radiatively induced
Chern-Simons term in a Lorentz- and CPT-violating modification of QED. Although
the coefficient of the induced Chern-Simons term is in general undetermined,
the nonperturbative theory appears to generate a definite value. However, the
CPT-even radiative corrections in this same formulation of the theory generally
break gauge invariance. We show that gauge invariance may yet be preserved
through the use of a Pauli-Villars regulator, and, contrary to earlier
expectations, this regulator does not necessarily give rise to a vanishing
Chern-Simons term. Instead, two possible values of the Chern-Simons coefficient
are allowed, one zero and one nonzero. This formulation of the theory therefore
allows the coefficient to vanish naturally, in agreement with experimental
observations.
