Protected states in $mbox{AdS}_3$ backgrounds from integrability

We write down the Algebraic Bethe Ansatz for string theory on $mbox{AdS}_3timesmbox{S}^3timesmbox{T}^4$ and $mbox{AdS}_3timesmbox{S}^3timesmbox{K3}$ 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 $mbox{AdS}_3timesmbox{S}^3timesmbox{S}^3times mbox{S}^1$ 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.