A Construction of non-degenerate $\mathbb{Z}_{2}$-harmonic functions on $\mathbb{R}^{n}$

We discover an explicit construction of non-degenerate $\mathbb{Z}_{2}$-harmonic functions on $\mathbb{R}^{n},n\geq 3$, using a variant of ellipsoidal coordinates on $\mathbb{R}^{n}$. The branching set of these examples is a codimension-$2$ ellipsoid, providing the first known family of non-degenerate $\mathbb{Z}_{2}$-harmonic $1$-forms on $\mathbb{R}^{n}$ with compact branching sets. Moreover, the graph of the related $\mathbb{Z}_{2}$-harmonic one form in $T^{*}\mathbb{R}^{n}$ can be obtained as a certain limit of a specific sequence of Lawlor's necks in $\mathbb{C}^{n}=T^{*}\mathbb{R}^{n}$.