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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 itto determine the wave-functions of protected closed strings in thesebackgrounds and prove that their energies are protected to all orders in$\alpha'$. We further apply the ABA to find the wave functions of protectedstates of $\mbox{AdS}_3\times\mbox{S}^3\times\mbox{S}^3\times \mbox{S}^1$ andits $\mathbf{Z}_2$ orbifold. Our findings match with protected spectrumcalculations from supergravity, $\mbox{Sym}^N$ orbifolds and apply to thecomplete moduli space of these theories, excluding orbifold blow-up modes forwhich further analysis is necessary.

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