  We discuss the static spherically symmetric Einstein-spinor field system in
the possible presence of various spinor field nonlinearities. We take into
account that the spinor field energy-momentum tensor (EMT) has in general some
off-diagonal components, whose vanishing due to the Einstein equations
substantially affects the form of the spinor field itself and the space-time
geometry.
  In particular, the EMT structure with any spinor field nonlinearities turns
out to be the same as that of the EMT of a minimally coupled scalar field with
a self-interaction potential. Therefore many results previously obtained for
systems with such scalar fields are directly extended to the Einstein-spinor
field system. Some special solutions are obtained and discussed, in particular,
a solution for the Einstein-Dirac system (which lack asymptotic flatness) and
some examples with spinor field nonlinearities.
