  We consider the Einstein-Dirac field equations describing a self-gravitating
massive neutrino, looking for axially-symmetric exact solutions; in the search
of general solutions, we find some that are specific and which have critical
features, such as the fact that the space-time curvature turns out to be flat
and the spinor field gives rise to a vanishing bi-linear scalar
$\overline{\psi}\psi=0$ with non-vanishing bi-linear pseudo-scalar
$i\overline{\psi}\gamma^5\psi\not=0$: because in quantum field theory general
computational methods are built on plane-wave solutions, for which bi-linear
pseudo-scalar vanishes while the bi-linear scalar does not vanish, then the
solutions we found cannot be treated with the usual machinery of quantum field
theory. This means that for the Einstein-Dirac system there exist admissible
solutions which nevertheless cannot be quantized with the common prescriptions;
we regard this situation as yet another issue of tension between Einstein
gravity and quantum principles. Possible ways to quench this tension can be
seen either in enlarging the validity of quantum field theory or by restricting
the space of the solutions of the Einstein-Dirac system of field equations.
