  This paper studies necessary conditions for the existence of alpha-surfaces
in complex space-time manifolds with nonvanishing torsion. 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. Interestingly, a particular solution of the
integrability condition is given by conformally right-flat and
right-torsion-free space-times.
