We study Einstein-Maxwell (non-null) source-free configurations that can be extended to any conformally invariant nonlinear electrodynamics (CINLE) by a constant rescaling of the electromagnetic field. We first obtain a criterion which characterizes such extendable solutions in terms either of the electromagnetic invariants, or (equivalently) of the canonical Newman-Penrose form of the self-dual Maxwell field. This is then used to argue that all static configurations are extendable (more generally, all configurations admitting a non-null twist-free Killing vector field). One can thus draw from the extensive literature to straightforwardly extend to CINLE various known exact solutions, whereby the duality invariance of the Einstein-Maxwell theory allows for dyonic solutions even in more general theories. This is illustrated by a few explicit examples, including the homogeneous $\mathrm{\Lambda }<0$ universe of Ozsváth, a black hole in the universe of Levi-Civita, Bertotti, and Robinson, a generalization of the charged $C$-metric, and nonexpanding gravitational waves in the Bonnor-Melvin background.

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• Classical black holes
• General relativity equations & solutions
• Classical electromagnetism