  In the framework of metric-free electrodynamics, we start with a {\em linear}
spacetime relation between the excitation 2-form $H = ({\cal D}, {\cal H})$ and
the field strength 2-form $F = ({E,B})$. This linear relation is constrained by
the so-called closure relation. We solve this system algebraically and extend a
previous analysis such as to include also singular solutions. Using the
recently derived fourth order {\em Fresnel} equation describing the propagation
of electromagnetic waves in a general {\em linear} medium, we find that for all
solutions the fourth order surface reduces to a light cone. Therefrom we derive
the corresponding metric up to a conformal factor.
