Riemann problem for the Born-Infeld system without differential constraints

We consider the Born-Infeld system without differential constraints, which occurs as soon as the differential constraints are not satisfied at the initial time. In such a case, the Poynting vector is not a conservative variable and the technique of enlargement of systems cannot be applied. In one space dimension, the resulting system consists of five conservative equations for which only one Riemann invariant exists. It is fully linearly degenerate but neither strictly hyperbolic nor rich. Under a smallness condition on the initial datum of only one variable, we prove that the Riemann problem has a unique entropy solution having discontinuities with three separated speeds. It is surprising that the result holds even for initial data lying in non-hyperbolic regions.