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M-theory Solutions Invariant under $D(2,1;γ) \oplus D(2,1;γ)$

We simplify and extend the construction of half-BPS solutions to 11-dimensional supergravity, with isometry superalgebra D(2,1;γ) \oplus D(2,1;γ). Their space-time has the form AdS_3 x S^3 x S^3 warped over a Riemann surface Σ. It describes near-horizon geometries of M2 branes ending on, or intersecting with, M5 branes along a common string. The general solution to the BPS equations is specified by a reduced set of data (γ, h, G), where γis the real parameter of the isometry superalgebra, and h and G are functions on Σwhose differential equations and regularity conditions depend only on the sign of γ. The magnitude of γenters only through the map of h, G onto the supergravity fields, thereby promoting all solutions into families parametrized by |γ|. By analyzing the regularity conditions for the supergravity fields, we prove two general theorems: (i) that the only solution with a 2-dimensional CFT dual is AdS_3 x S^3 x S^3 x R^2, modulo discrete identifications of the flat R^2, and (ii) that solutions with γ< 0 cannot have more than one asymptotic higher-dimensional AdS region. We classify the allowed singularities of h and G near the boundary of Σ, and identify four local solutions: asymptotic AdS_4/Z_2 or AdS_7&#39; regions; highly-curved M5-branes; and a coordinate singularity called the &#34;cap&#34;. By putting these &#34;Lego&#34; pieces together we recover all known global regular solutions with the above symmetry, including the self-dual strings on M5 for $γ< 0$, and the Janus solution for γ> 0, but now promoted to families parametrized by |γ|. We also construct exactly new regular solutions which are asymptotic to AdS_4/Z_2 for γ< 0, and conjecture that they are a different superconformal limit of the self-dual string.