  We prove in one go that each of the 4 families of Burniat surfaces with K^2 =
6,5,4, is a connected component of the moduli space of surfaces of general
type.
  We prove also the rationality of each component. In the nodal case (one of
the two families for K^2_S = 4) a very surprising and new phenomenon occurs.
  Both the moduli space for the minimal models and the Gieseker moduli space
for canonical models are everywhere non reduced.
  But the nilpotence order is higher for the first.
