  The complete Type IIB supergravity solutions with 16 supersymmetries are
obtained on the manifold $AdS_4 \times S^2 \times S^2 \times \Sigma$ with
$SO(2,3) \times SO(3) \times SO(3)$ symmetry in terms of two holomorphic
functions on a Riemann surface $\Sigma$, which generally has a boundary. This
is achieved by reducing the BPS equations using the above symmetry
requirements, proving that all solutions of the BPS equations solve the full
Type IIB supergravity field equations, mapping the BPS equations onto a new
integrable system akin to the Liouville and Sine-Gordon theories, and mapping
this integrable system to a linear equation which can be solved exactly.
Amongst the infinite class of solutions, a non-singular Janus solution is
identified which provides the AdS/CFT dual of the maximally supersymmetric
Yang-Mills interface theory discovered recently. The construction of general
classes of globally non-singular solutions, including fully back-reacted $AdS_5
\times S^5$ and supersymmetric Janus doped with D5 and/or NS5 branes, is
deferred to a companion paper.
