  We prove two theorems which imply that any stationary nonlinear
electromagnetic field obeying a dominant energy condition in a strictly
stationary, everywhere regular, asymptotically flat spacetime must be either
trivial or a stealth field. The first theorem holds in static spacetimes and is
independent of the gravitational part of the action, as long as the coupling of
the electromagnetic field to the gravitational field is minimal. The second
theorem assumes Einstein--Hilbert gravitational action and relies on the
positive energy theorem, but does not assume that the spacetime metric is
static. In addition, we discuss possible generalizations of these results, to
theories with charged matter, as well as higher-dimensional nonlinear
electromagnetic fields.
