Nonlinear electromagnetic fields in strictly stationary spacetimes

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