Doubly-fluctuating BPS solutions in six dimensions

We analyze the BPS solutions of minimal supergravity coupled to an anti-self-dual tensor multiplet in six dimensions and find solutions that fluctuate non-trivially as a function of two variables. We consider families of solutions coming from KKM monopoles fibered over Gibbons-Hawking metrics or, equivalently, non-trivial T-2 fibrations over an R-3 base. We find smooth microstate geometries that depend upon many functions of one variable, but each such function depends upon a different direction inside the T-2 so that the complete solution depends non-trivially upon the whole T-2. We comment on the implications of our results for the construction of a general superstratum.