Asymptotically flat gravitating spinor field solutions. Step 3 -   Einstein-Dirac equations: cylindric case
  Einstein-Dirac equations for two spinor fields are considered. It is shown that in this case one can obtain self-consistent equations set for these gravitating spinors. The key idea to obtain Einstein-Dirac equations is to use special ans\"atze for spinor fields in order to zero some off-diagonal componentes of the spinor energy - momentum tensor. 
Vladimir Dzhunushaliev
gr-qc