  Within the scope of Bianchi type-III spacetime we study the role of spinor
field on the evolution of the Universe as well as the influence of gravity on
the spinor field. In doing so we have considered a polynomial type of
nonlinearity. In this case the spacetime remains locally rotationally symmetric
and anisotropic all the time. It is found that depending on the sign of
nonlinearity the models allows both accelerated and oscillatory modes of
expansion. The non-diagonal components of energy-momentum tensor though impose
some restrictions on metric functions and components of spinor field, unlike
Bianchi type I, V and $VI_0$ cases, they do not lead to vanishing mass and
nonlinear terms of the spinor field.
