  In this paper we study the geometry of the total space $Y$ of a cotangent
bundle to a K\"ahler manifold $N$ where $N$ is obtained as a K\"ahler reduction
from $\mathbb C^n$. Using the hyperk\"ahler reduction we construct a
hyperk\"ahler metric on $Y$ and prove that it coincides with the canonical
Feix-Kaledin metric. This metric is in general non-complete. We show that the
metric completion $\tilde Y$ of the space $Y$ is equipped with a structure of a
stratified hyperk\"ahler space. We give a necessary condition for the
Feix-Kaledin metric to be complete using an observation of R.Bielawski. Pick a
complex structure $J$ on $\tilde Y$ induced from quaternions. Suppose that
$J\ne\pm I$ where $I$ is the complex structure whose restriction to $Y = T^*N$
is induced by the complex structure on $N$. We prove that the space
$\tilde{Y}_J$ admits an algebraic structure and is an affine variety.
