  We study heterotic supergravity on the conifold and its Z2 orbifold with
Abelian gauge fields and three-form flux. At large distances, these solutions
are locally Ricci-flat, have a magnetic flux through the two-sphere at infinity
as well as non-zero five-brane charge. For a given flux, our family of
solutions has three real parameters, the size of the pair of two spheres in the
IR and the dilaton zero mode. We present an explicit analytic solution for the
decoupled near horizon region where for a given flux, the size of the cycles is
frozen and the only parameter is the dilaton zero mode. We also present an
exactly solvable worldsheet CFT for this near horizon region. When one of the
two cycles has vanishing size, the near horizon region no longer exists but we
obtain a solution on the (unorbifolded) resolved conifold.
