The fact that the double cover of the Atiyah–Hitchin manifold admits a $3$–parameter family of $D_{1}$ ALF deformations can also be shown using methods similar to the ones developed in this paper. Indeed, it is known [24, §5.4] that the rotationally invariant $D_{1}$ ALF metric admits a unique $L^{2}$–integrable (in fact, exponentially decaying) anti-self-dual harmonic form $\eta$. This form yields a $3$–dimensional space of infinitesimal hyperkähler deformations and an extension of the analysis needed for the proof of Theorem 6.15 could be used to integrate these infinitesimal deformations to genuine $D_{1}$ ALF metrics. In fact Dancer [14] has constructed a $3$–parameter family of hyperkähler deformations of the rotationally invariant $D_{1}$ ALF metric using Nahm’s equations and hyperkähler quotient techniques: there exists a hyperkähler $8$–manifold $\mathcal{N}$ constructed as a moduli space of solutions to Nahm’s equations which admits a triholomorphic $U(1)$–action. Denote by $\mu\colon\thinspace\mathcal{N}\rightarrow\mathbb{R}^{3}$ the corresponding hyperkähler moment map. Dancer identifies the rotationally symmetric $D_{1}$ ALF metric with the hyperkähler quotient $\mu^{-1}(0)/U(1)$. By varying the level set of the moment map he then obtains a $3$–parameter family of hyperkähler deformations of the Atiyah–Hitchin metric. By a general formula for the infinitesimal deformation of the symplectic form of a symplectic quotient corresponding to varying the level set of the moment map [17], the infinitesimal deformations of the Atiyah–Hitchin metric corresponding to Dancer’s metrics coincide with those determined by the $L^{2}$ harmonic form $\eta$, which is interpreted in this context as the curvature of the natural hyperholomorphic connection on the $U(1)$–bundle $\mu^{-1}(0)\rightarrow\mu^{-1}(0)/U(1)$ induced by the Levi–Civita connection of $\mathcal{N}$.