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.