  We show that the solutions to the equations defining the so-called Calabi-Yau
condition for fourth order operators of degree two defines a variety that
consists of ten irreducible components. These can be described completely in
parametric form, but only two of the components seem to admit arithmetically
interesting operators. We include a description of the 69 essentially distinct
fourth order Calabi-Yau operators of degree two that are presently known to us.
