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Quarter-BPS AdS5 solutions in M-theory with a T2 bundle over a Riemann surface

We study and classify quarter-BPS AdS5 systems in M-theory, whose internal six-dimensional geometry is a T2 bundle over a Riemann surface and two interval directions. The general system presented, provides a unified description of all known AdS5 solutions in M-theory. These systems are governed by two functions, one that corresponds to the conformal factor of the Riemann surface and another that describes the T2 fibration. We find solutions that can be organized into two classes. In the first one, solutions are specified by the conformal factor of the Riemann surface which satisfies a warped generalization of the SU(infinity) Toda equation. The system in the second class requires the Riemann surface to be S2, H2 or T2. Class one contains the M-theory AdS5 solutions of Lin, Lunin and Maldacena; the solutions of Maldacena and Nunez; the solutions of Gauntlett, Martelli, Sparks and Waldram; and the eleven-dimensional uplift of the Y(p,q) metrics. The second includes the recently found solutions of Beem, Bobev, Wecht and the author. Within each class there are new solutions that will be studied in a companion paper.