  We simplify and extend the construction of half-BPS solutions to
11-dimensional supergravity, with isometry superalgebra D(2,1;\gamma) \oplus
D(2,1;\gamma). Their space-time has the form AdS_3 x S^3 x S^3 warped over a
Riemann surface \Sigma. 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 (\gamma, h,
G), where \gamma is the real parameter of the isometry superalgebra, and h and
G are functions on \Sigma whose differential equations and regularity
conditions depend only on the sign of \gamma. The magnitude of \gamma enters
only through the map of h, G onto the supergravity fields, thereby promoting
all solutions into families parametrized by |\gamma|. 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 \gamma < 0 cannot have more than one asymptotic
higher-dimensional AdS region. We classify the allowed singularities of h and G
near the boundary of \Sigma, and identify four local solutions: asymptotic
AdS_4/Z_2 or AdS_7' regions; highly-curved M5-branes; and a coordinate
singularity called the "cap". By putting these "Lego" pieces together we
recover all known global regular solutions with the above symmetry, including
the self-dual strings on M5 for $\gamma < 0$, and the Janus solution for \gamma
> 0, but now promoted to families parametrized by |\gamma|. We also construct
exactly new regular solutions which are asymptotic to AdS_4/Z_2 for \gamma < 0,
and conjecture that they are a different superconformal limit of the self-dual
string.
