  We give an explicit construction and clasification of some very special sort
of Enriques surfaces in characteristic two. This proves the existence of some
of the surfaces that were called ``extra-special'' by Cossec and Dolgachev in
their book on the subject, and whose existence was left open by them. These
surfaces coincide with a class of surfaces that are introduced in recent work
by Ekedahl and Shepherd--Barron.
