On degraded two message set broadcasting

We consider the two message set problem, where a source broadcasts a common message W
₁
to an arbitrary set of receivers U and a private message W
₂
to a subset of the receivers P ¿ U. Transmissions occur over linear deterministic channels. For the case where at most two receivers do not require the private message, we give an exact characterization of the capacity region, where achievability is through linear coding.