  We investigate self-gravitating solutions of the Einstein-Skyrme theory
coupled to spin-isospin Dirac fermions and consider the dependence of the
spectral flow on the effective gravitational coupling constant and on the
Yukawa coupling. It is shown that the effects of the backreaction of the
fermionic mode may strongly deform the configuration. Depending on the choice
of parameters, solutions with positive, negative and zero ADM mass may arise.
The occurrence of regular anti-gravitating asymptotically flat solutions with
negative ADM mass is caused by the violation of the energy conditions.
