With the notation as above, let $h(y)=f(y_{1},y_{2})+ig(y_{1},y_{2})$ be a holomorphic function on $D_{r}$, so that $-{1\over 4\pi}\log|y|^{2}+f(y_{1},y_{2})>0$ on $D_{r}$. Let $V_{0}$ be the harmonic function on $Y$ defined in Lemma 3.1, and $V=V_{0}+f(y_{1},y_{2})/\epsilon$, with $\epsilon$ chosen small enough so that $V>0$ on $Y$. Then there exists a connection 1-form $\theta$ on $X$ such that $d\theta/2\pi i=*dV$, and this defines a hyperkähler metric on $X$ with

$$\eqalign{-\mathop{\rm Re}\Omega&=dy_{1}\wedge\theta/2\pi i+Vdy_{2}\wedge du\cr-\mathop{\rm Im}\Omega&=dy_{2}\wedge\theta/2\pi i+Vdu\wedge dy_{1}\cr\omega&=du\wedge\theta/2\pi i+Vdy_{1}\wedge dy_{2}.\cr}$$

These forms extend to $\bar{X}$, giving a hyperkähler metric on $\bar{X}$, and a holomorphic elliptic fibration $\bar{X}\rightarrow D_{r}$ with periods $1$ and ${1\over 2\pi i}\log y+ih(y)+C$, for some real constant $C$. By appropriate choice of $\theta$, this constant $C$ may be taken to be zero.