  We consider the single-handed spinor field in interaction with its own
gravitational field described by the set of field equations given by Weyl field
equations written in terms of derivatives that are covariant with respect to
the gravitational connection plus Einstein field equations soured with the
energy tensor of the spinor: for the Weyl spinor and the ensuing spacetime of
Weyl-Lewis-Papapetrou structure, we will find all exact solutions. The obtained
solution for the metric tensor is that of a PP-wave spacetime while the spinor
field is a flag-dipole.
