We prove the uniqueness of the supersymmetric Salam-Sezgin (Minkowski) 4 ×S 2 ground state among all nonsingular solutions with a four-dimensional Poincaré, de Sitter or anti-de Sitter symmetry. We construct the most general solutions with an axial symmetry in the two-dimensional internal space, and show that included amongst these is a family that is non-singular away from a conical defect at one pole of a distorted 2-sphere. These solutions admit the interpretation of 3-branes with negative tension.

The Penrose limit is generalized to show that, any leading order solution of the low-energy field equations in any one of the five string theories has a plane wave solution as a limit. This limiting procedure takes into account all the massless fields that may arise and commutes with the T-duality so that any dual solution has again a plane wave limit. The scaling rules used in the limit are unique and stem from the scaling property of the D = 11 supergravity action. Although the leading order dual solutions need not be exact or supersymmetric, their plane wave limits always preserve some portion of the Poincaré supersymmetry and solve the relevant field equations in all powers of the string tension parameter. Further properties of the limiting procedure are discussed.

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We find a new family of supersymmetric vacuum solutions in the six-dimensional chiral gauged N = (1, 0) supergravity theory. They are generically of the form AdS 3 × S 3 , where the 3-sphere is squashed homogeneously along its Hopf fibres. The squashing is freely adjustable, corresponding to changing the 3-form charge, and the solution is supersymmetric for all squashings. In a limit where the length of the Hopf fibres goes to zero, one recovers, after a compensating rescaling of the fibre coordinate, a solution that is locally the same as the well-known (Minkowski) 4 × S 2 vacuum of this theory. It can now be viewed as a fine tuning of the new more general family. The traditional "Cosmological Constant Problem" is replaced in this theory by the problem of why the four-dimensional (Minkowski) 4 × S 2 vacuum should be selected over other members of the equally supersymmetric AdS 3 × S 3 family. We also obtain a family of dyonic string solutions in the gauged N = (1, 0) theory, whose near-horizon limits approach the AdS 3 times squashed S 3 solutions.