Axially symmetric exact solutions for flagpole fermions with gravity

The Lounesto classification splits spinors into six classes: I, II, III are those for which at least one among scalar and pseudo-scalar bilinear 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 bilinear spinor quantities are identically zero, its spinors are called singular, and they are split into 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 themomentum 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 (Cianci et al., Eur Phys J C 76:595, 2016), and Weyl fields (Cianci et al., Eur Phys J C 75:478, 2015), 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.