  We analyze the structure of simply-connected Enriques surface in
characteristic two whose K3-like covering is normal, building on the work of
Ekedahl, Hyland and Shepherd-Barron. We develop general methods to construct
such surfaces and the resulting twistor lines in the moduli stack of Enriques
surfaces, including the case that the K3-like covering is a normal rational
surface rather then a normal K3 surface. Among other things, we show that
elliptic double points indeed do occur. In this case,there is only one
singularity.The main idea is to apply flops to Frobenius pullbacks of rational
elliptic surfaces, to get the desired K3-like covering. Our results hinge on
Lang's classification of rational elliptic surfaces, the determination of their
Mordell--Weil lattices by Shioda and Oguiso, and the behavior of unstable
fibers under Frobenius pullback via Ogg's Formula. Along the way, we develop a
general theory of Zariski singularities in arbitrary dimension, which is
tightly interwoven with the theory of height-one group schemes actions and
restricted Lie algebras. Furthermore, we determine under what conditions
tangent sheaves are locally free, and introduce a theory of canonical coverings
for arbitrary proper algebraic schemes.
