  A CY bundle on a connected compact complex manifold $X$ was a crucial
ingredient in constructing differential systems for period integrals in [LY],
by lifting line bundles from the base $X$ to the total space. A question was
therefore raised as to whether there exists such a bundle that supports the
liftings of all line bundles from $X$, simultaneously. This was a key step for
giving a uniform construction of differential systems for arbitrary complete
intersections in $X$. In this paper, we answer the existence question in the
affirmative if $X$ is assumed to be K\"ahler, and also in general if the Picard
group of $X$ is assumed to be discrete. Furthermore, we prove a rigidity
property of CY bundles if the principal group is an algebraic torus, showing
that such a CY bundle is essentially determined by its character map.
