  We systematically study all supersymmetric solutions of six-dimensional (2,0)
supergravity with a null isometry. In particular, every such solution with at
least four real supersymmetries is also a supersymmetric solution of a (1,0)
theory preserving the same absolute amount of supersymmetry. This implies that
no genuinely new solutions of this type can be found in this framework. The
microstate geometries associated to supersymmetric black holes within Mathur's
proposal are generically supersymmetric solutions of six-dimensional
supergravity. A direct consequence of our result is that supersymmetric
microstate geometries of single centre supersymmetric black holes should carry
only one compact 3-cycle.
