  Recently, a metric construction for the Calabi-Yau 3-folds from a
four-dimensional hyperkahler space by adding a complex line bundle was
proposed. We extend the construction by adding a U(1) factor to the holomorphic
(3,0)-form, and obtain the explicit formalism for a generic hyperkahler base.
We find that a discrete choice arises: the U(1) factor can either depend solely
on the fibre coordinates or vanish. In each case, the metric is determined by
one differential equation for the modified Kahler potential. As explicit
examples, we obtain the generalized resolutions (up to orbifold singularity) of
the cone of the Einstein-Sasaki spaces Y^{p,q}. We also obtain a large class of
new singular CY3 metrics with SU(2)\times U(1) or SU(2)\times U(1)^2
isometries.
