  The extreme magnetoelectric medium (EME medium) is defined in terms of two
medium dyadics, $\alpha$, producing electric polarization by the magnetic field
and $\beta$, producing magnetic polarization by the electric field. Plane-wave
propagation of time-harmonic fields of fixed finite frequency in the EME medium
is studied. It is shown that (if $\omega\neq 0$) the dispersion equation has a
cubic and homogeneous form, whence the wave vector $k$ is either zero or has
arbitrary magnitude. In many cases there is no dispersion equation ("NDE
medium") to restrict the wave vector in an EME medium. Attention is paid to the
case where the two medium dyadics have the same set of eigenvectors. In such a
case the $k$ vector is restricted to three eigenplanes defined by the medium
dyadics. The emergence of such a result is demonstrated by considering a more
regular medium, and taking the limit of zero permittivity and permeability. The
special case of uniaxial EME medium is studied in detail. It is shown that an
interface of a uniaxial EME medium appears as a DB boundary when the axis of
the medium is normal to the interface. More in general, EME media display
interesting wave effects that can potentially be realized through metasurface
engineering.
