Nonlinear duality-invariant conformal extension of Maxwell's equations

All nonlinear extensions of the source-free Maxwell equations preserving both SO(2) electromagnetic duality invariance and conformal invariance are found, and shown to be limits of a one-parameter generalization of Born-Infeld electrodynamics. The strong-field limit is the same as that found by Bialynicki-Birula from Born-Infeld theory but the weak-field limit is a new one-parameter extension of Maxwell electrodynamics, which is interacting but admits exact light-velocity plane-wave solutions of arbitrary polarization. Small-amplitude waves on a constant uniform electromagnetic background exhibit birefringence, but one polarization mode remains lightlike.