  The Lounesto classification splits spinors in six classes: I, II, III are
those for which at least one among scalar and pseudo-scalar bi-linear spinor
quantities is non-zero, its spinors are called regular, and among them we find
the usual Dirac spinor. IV, V, VI are those for which the scalar and
pseudo-scalar bi-linear spinor quantities are identically zero, its spinors are
called singular, and they are split in further sub-classes: IV has no further
restrictions, its spinors are called flag-dipole; V is the one for which the
spin axial-vector vanishes, its spinors are called flagpole, and among them we
find the Majorana spinor; VI is the one for which the momentum
antisymmetric-tensor vanishes, its spinors are called dipole, and among them we
find the Weyl spinor. In the quest for exact solutions of fully-coupled systems
of spinor fields in their own gravity, we have already given examples in the
case of Dirac fields and Weyl fields but never in the case of Majorana or more
generally flagpole spinor fields. Flagpole spinor fields in interaction with
their own gravitational field, in the case of axial symmetry, will be
considered. Exact solutions of the field equations will be given.
