Equations for some very special Enriques surfaces in characteristic two

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.