Let $U=\mathbb{R}^{3}$. If $V=\sigma$ (a positive constant), then $(\mathfrak{M},g)$ is a flat product $\mathbb{R}^{3}\times S^{1}$. If $V=\frac{1}{2r}$, then $(\mathfrak{M},g)$ is flat Euclidean space $\mathbb{R}^{4}$ and the map $\pi$ is exactly the Hopf fibration. If $V_{\sigma}=\sigma+\frac{1}{2r}$ then $(\mathfrak{M},g)$ is the Taub-NUT space. This is again diffeomorphic to $\mathbb{R}^{4}$ but has cubic volume growth and is asymptotic to an $S^{1}$ fibration over $\mathbb{R}^{3}\setminus K$ at infinity where the length of the $S^{1}$ fibers approaches a positive constant. Notice that as $\sigma$ varies, these metrics are isometric up to dilation. This is most easily seen using the above intrinsic description. We take the metric $g_{1}$ constructed using $V_{1}=1+\frac{1}{2r}$ and rescale $g_{\sigma}=\sigma^{-1}g_{1}$. Then the length of the $S^{1}$ orbits becomes $(\sigma V_{1})^{-1/2}$ and the hyperkähler moment map becomes $\pi_{\sigma}=\sigma^{-1}\pi$. Thus, $g_{\sigma}$ can be written in Gibbons-Hawking form with potential $\sigma V_{1}=\sigma+\frac{1}{2r_{\sigma}}$. See Lemma 7.9 for more details.

By taking multiple poles, we similarly obtain other hyperkähler manifolds which are asymptotic to quotients of either $\mathbb{R}^{4}$ or Taub-NUT space by cyclic groups. These are usually referred to in the literature as ALE and ALF spaces of $A_{k-1}$ type. In particular, see [Min11] for a complete theory of ALF-$A_{k-1}$ spaces.