Noether gauge symmetry of Dirac field in $2+1$ dimensional gravity

In this paper, we consider a gravitational theory including a Dirac field that is non-minimal coupled to gravity in $2+1$ dimensions. Noether gauge symmetry approach can be used to fix the form of coupling function $F(\Psi)$ and the potential $V(\Psi)$ of the Dirac field and to obtain a constant of motion for the dynamical equations. In the context of $2+1$ dimensions gravity, we investigate cosmological solutions of the field equations using these forms obtained by the existent of Noether gauge symmetry. In this picture, it is shown that for the non-minimal coupling case, the cosmological solutions indicate both an early-time inflation and late-time acceleration for the universe.