  We find a large new class of explicit solutions preserving four supercharges
in six-dimensional supergravity. The solutions are determined by solving a
linear system of equations on a four-dimensional Kahler base studied by LeBrun.
For particular choices of the parameters, we find regular backgrounds that are
asymptotic to the near-horizon limit of the D1-D5-P black string. Holography
implies that these backgrounds should be dual to 1/4-BPS states in the D1-D5-P
CFT.
