  The necessary and sufficient condition for the existence of $\alpha$-surfaces
in complex space-time manifolds with nonvanishing torsion is derived. For these
manifolds, Lie brackets of vector fields and spinor Ricci identities contain
explicitly the effects of torsion. This leads to an integrability condition for
$\alpha$-surfaces which does not involve just the self-dual Weyl spinor, as in
complex general relativity, but also the torsion spinor, in a nonlinear way,
and its covariant derivative. A similar result also holds for four-dimensional,
smooth real manifolds with a positive-definite metric. Interestingly, a
particular solution of the integrability condition is given by conformally
right-flat and right-torsion-free space-times.
