  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 R3 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.
