  We write down the Algebraic Bethe Ansatz for string theory on
$\mbox{AdS}_3\times\mbox{S}^3\times\mbox{T}^4$ and
$\mbox{AdS}_3\times\mbox{S}^3\times\mbox{K3}$ in its orbifold limits. We use it
to determine the wave-functions of protected closed strings in these
backgrounds and prove that their energies are protected to all orders in
$\alpha'$. We further apply the ABA to find the wave functions of protected
states of $\mbox{AdS}_3\times\mbox{S}^3\times\mbox{S}^3\times \mbox{S}^1$ and
its $\mathbf{Z}_2$ orbifold. Our findings match with protected spectrum
calculations from supergravity, $\mbox{Sym}^N$ orbifolds and apply to the
complete moduli space of these theories, excluding orbifold blow-up modes for
which further analysis is necessary.
