  We consider the problem of finding exact, eleven-dimensional, BPS
supergravity solutions in which the compactification involves a non-trivial
Calabi-Yau manifold, ${\cal Y}$, as opposed to simply a $T^6$. Since there are
no explicitly-known metrics on non-trivial, compact Calabi-Yau manifolds, we
use a non-compact "local model" and take the compactification manifold to be
${\cal Y} = {\cal M}_{GH} \times T^2$ where ${\cal M}_{GH}$ is a
hyper-K\"ahler, Gibbons-Hawking ALE space. We focus on backgrounds with three
electric charges in five dimensions and find exact families of solutions to the
BPS equations that have the same four supersymmetries as the three-charge black
hole. Our exact solution to the BPS system requires that the Calabi-Yau
manifold be fibered over the space-time using compensators on ${\cal Y}$. The
role of the compensators is to ensure smoothness of the eleven-dimensional
metric when the moduli of ${\cal Y}$ depend on the space-time. The Maxwell
field Ansatz also implicitly involves the compensators through the frames of
the fibration. We examine the equations of motion and discuss the brane
distributions on generic internal manifolds that do not have enough symmetry to
allow smearing.
