  The subject of this paper is the explicit momentum construction of complete
Einstein metrics by ODE methods. Using the Calabi ansatz, further generalized
by Hwang-Singer, we show that there are non-trivial complete conformally
K\"ahler Einstein metrics on certain Hermitian holomorphic vector bundles and
their subbundles over complete K\"ahler-Einstein manifolds. In special cases,
we give the explicit expressions of of these metrics. These examples show that
there is a compact K\"ahler manifold $M$ and its subvariety $N$ whose
codimension is greater than 1 such that there is a complete conformally
K\"ahler Einstein metric on $M-N$.
