  The specific behavior of spinor field in curve space-time with the exception
of FRW model almost always gives rise to non-trivial non-diagonal components of
the energy-momentum tensor. This non-triviality of non-diagonal components of
the energy-momentum tensor imposes some severe restrictions either on the
spinor field or on the metric functions. In this paper within the scope of an
anisotropic Bianchi type-I Universe we study the role of spinor field in the
evolution of the Universe. It is found that there exist two possibilities. In
one scenario the initially anisotropic Universe evolves into an isotropic one
asymptotically, but in this case the spinor field itself undergoes some severe
restrictions. In the second scenario the isotropization takes places almost at
the beginning of the process.
