  We find classical closed string solutions to the non-relativistic
AdS$_5\times$S$^5$ string theory which are the analogue of the BMN and GKP
solutions for the relativistic theory. We show that non-relativistic
AdS$_5\times$S$^5$ string theory admits a $\mathbb{Z}_2$ orbifold symmetry
which allows us to impose twisted boundary conditions. Among the solutions in
the twisted sector, we find the one around which the semiclassical expansion in
arXiv:2102.00008 takes place.
