Non-integrability in AdS$_3$ vacua

We ask the question of classical integrability for certain (classes of) supergravity vacua that contain an AdS$_3$ factor arising in massive IIA and IIB theories and realizing various and different amounts of supersymmetry. Our approach is based on a well-established method of analytic non-integrability for Hamiltonian systems. To detect a non-integrable sector we consider a non-trivially wrapped string soliton and study its fluctuations. We answer in the negative for each and every one of the supergravity solutions. That is, of course, modulo very specific limits where the metrics reduce to the AdS$_3 \times$ S$^3 \times \tilde{S}^3 \times$ S$^1$ and AdS$_3 \times$ S$^3 \times$ T$^4$ solutions which are known to be integrable.