On exceptional Enriques surfaces
  We give a complete description of all classical Enriques surfaces with non-zero global vector fields. In particular we show that there are such surfaces. The obtained result also applies to supersingular Enriques surfaces fulfilling a rather special condition. During this classification we study some properties of genus 1-fibrations special to characteristic as well as make a close study of the genus 1-fibration on the surfaces that we classify. 
T. Ekedahl, N. I. Shepherd-Barron