  Using Hamiltonian methods, we find six relativistic theories of nonlinear
electrodynamics for which plane wave perturbations about a constant uniform
background are not birefringent. All have the same conformal strong-field limit
to Bialynicki-Birula (BB) electrodynamics, but only four avoid superluminal
propagation: Born-Infeld (BI), its non-conformal ``extreme'' limits (electric
and magnetic) and the conformal BB limit. The quadratic dispersion relation of
BI is shown to degenerate in the extreme limits to a pair of linear relations,
which become identical in the BB limit.
