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.
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