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It is known that string theory on $AdS_3\times M_7$ backgrounds preserving 16supercharges is classically integrable. This integrability has been previouslyused to write down a set of integral equations, known as the finite-gapequations. These equations can be solved for the closed string spectrum of thetheory. However, it has been known for some time that the $AdS_3\times M_7$finite-gap equations do not capture the dynamics of the massless modes of theclosed string theory. In this paper we re-examine the derivation of the$AdS_3\times M_7$ finite-gap system. We find that the conditions that hadpreviously been imposed on these integral equations in order to implement theVirasoro constraints are too strict, and are in fact not required. We identifythe correct implementation of the Virasoro constraints on finite-gap equationsand show that this new, less restrictive condition captures the complete closedstring spectrum on $AdS_3\times M_7$.

$AdS_3/CFT_2$, finite-gap equations and massless modes