Let $u,v:\mathbin{\mathbb{R}}^{2}\rightarrow\mathbin{\mathbb{R}}$ be solutions of (33). Then there exists a unique function $f:\mathbin{\mathbb{R}}^{2}\rightarrow\mathbin{\mathbb{R}}$ with

$$\begin{gathered}\frac{\partial f}{\partial x}=u,\quad\frac{\partial f}{\partial y}=v\quad\text{and}\quad f(0,0)=0,\\ \text{satisfying}\quad\frac{\partial^{2}f}{\partial x^{2}}+\Bigl(4\Bigl(\frac{\partial f}{\partial x}\Bigr)^{2}+4y^{2}+a^{2}\Bigr)^{1/2}\frac{\partial^{2}f}{\partial y^{2}}=0.\end{gathered}$$

(39)

Conversely, all solutions of (39) yield solutions of (33).