  We find supergravity solutions corresponding to all U(1) x U(1) invariant
chiral primaries of the D1-D5-KK system. These solutions are 1/8 BPS, carry
angular momentum, and are asymptotically flat in the 3+1 dimensional sense.
They can be thought of as representing the ground states of the four
dimensional black hole constructed from the D1-D5-KK-P system. Demanding the
absence of unphysical singularities in our solutions determines all free
parameters, and gives precise agreement with the quantum numbers expected from
the CFT point of view. The physical mechanism behind the smoothness of the
solutions is that the D1-branes and D5-branes expand into a KK-monopole
supertube in the transverse space of the original KK-monopole.
