The gluing construction presented in this paper could be extended to the non-compact setting to yield yet another construction of dihedral ALF metrics. Indeed, one considers a Gibbons–Hawking metric obtained from the harmonic function

$$h=\lambda-\frac{2}{|x|}+\sum_{i=1}^{m}{\frac{1}{2|x-x_{i}|}+\frac{1}{2|x+x_{i}|}}$$

for $m$ distinct points $x_{1},\dots,x_{m}\in\mathbb{R}^{3}\setminus\{0\}$. Observe that for $\lambda>0$ sufficiently large $h>0$ outside an arbitrarily small neighbourhood of the origin. Since the configuration of punctures is invariant under the standard involution of $\mathbb{R}^{3}$, this (incomplete) metric descends to a hyperkähler metric on a $\mathbb{Z}_{2}$ quotient. For $\lambda$ sufficiently large one can then complete this metric by gluing in a copy of the $D_{0}$ ALF space close to the origin. This approximate solution could then be deformed to an exact hyperkähler metric in a way similar to the proof of Theorem 6.15.