  We demonstrate a solution generating technique, modulo some constraints, for
a large class of smooth supergravity solutions with the same asymptotic charges
as a five dimensional 3-charge BPS black hole or black ring, dual to a D1/D5/P
system. These solutions are characterized by a harmonic function with both
positive and negative poles, which induces a geometric transition whereby
singular sources have disappeared and all of the net charge at infinity is
sourced by fluxes through two-cycles joining the poles of the harmonic
function.
