  We study surfaces of bidegree (1,d) contained in the flag threefold in
relation to the twistor projection. In particular, we focus on the number and
the arrangement of twistor fibers contained in such surfaces. First, we prove
that there is no irreducible surface of bidegree (1,d) containing d+2 twistor
fibers in general position. On the other hand, given any collection of (d+1)
twistor fibers satisfying a mild natural constraint, we prove the existence of
a surface of bidegree (1,d) that contains them. We improve our results for d=2
or d=3, by removing all the generality hypotheses.
