  We show that Hilbert schemes of points on supersingular Enriques surface in
characteristic 2 are simply connected, symplectic varieties but are not
irreducible symplectic as the hodge number $h^{2,0} > 1$, even though a
supersingular Enriques surface is an irreducible symplectic variety. These are
the classes of varieties which appear only in characteristic 2 and they show
that the hodge number formula for G\"ottsche-Soergel does not hold over
characteristic 2. It also gives examples of varieties with trivial canonical
class which are neither irreducible symplectic nor Calabi-Yau, thereby showing
that there are strictly more classes of simply connected varieties with trivial
canonical class in characteristic 2 than over $\mathbb{C}$ as given by
Beauville-Bogolomov decomposition theorem.
