  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 generalisation 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 polarisation.
Small-amplitude waves on a constant uniform electromagnetic background exhibit
birefringence, but one polarisation mode remains lightlike.
