Let $\mathbb{S}^{2N - 1}$ be the unit sphere in $\mathbb{C}^N$ and let $L$ be the Lewy hypersurface, which is defined by $$ \operatorname{Im} Z_N = \sum_{j = 0}^{N - 1} |Z_j|^2. $$ Let $H(\mathbf{Z}):\mathbb{S}^{2N - 1} \to L$ be a holomorphic rational mapping $ H(\mathbf{Z}) = (H_1(\mathbf{Z}), \dots, H_N(\mathbf{Z})) $ defined by $$ H_j(\mathbf{Z}) = \frac{iZ_j}{1 - Z_N}$$ for $j = 1, \dots, N - 1$, and $$ H_N(\mathbf{Z}) = \frac{i(Z_N + 1)}{1 - Z_N}. $$ Prove that $H$ is a bijection.
The problem statement is as above. This is from a "reader should easily be able to check" comment in Baouend, Ebenfelt, and Rothschild's Real Submanifolds in Complex Space and their Mappings. Here is my attempt:
We show that $H$ is surjective. Let $\mathbf{U} = (U_1,\dots,U_N) = (U_1,\dots,V_N + iW_N) \in L \subset \mathbb{C}^N$. Then by the definition of the Lewy hypersurface, we compute: $$ V_N + iW_N = V_N + i \sum_{j = 1}^{N - 1} |U_j|^2 = \frac{i(Z_N + 1)}{1 - Z_N}. $$
I'm a bit stuck at this point. This doesn't seem terribly complicated, maybe just algebraic manipulation. But it doesn't seem like there's a nice way to get an expression for $\mathbf{Z}$ in terms of the components of $\mathbf{U}$ such that $\mathbf{U} \mapsto \mathbf{Z}$. We also know $\mathbf{Z}$ is on the sphere, but I don't immediately see how we can use that.