This question is related to this question: "Solutions of equations characterizing a complex structure." Where, in this case we suppose the Euclidean space instead of Sphere.

Let $\mathbb{R}^{2n}$ be the Euclidean $2n$ dimensional space and $(x^1,...,x^n,y^1,...,y^n)$ be its coordinate system. Suppose $u=\frac{\beta}{\delta}$ and $v=\frac{1}{\delta}$ where $\beta , \delta$ are real-valued functions on $\mathbb{R}^{2n}$. Is there any other solutions except constant functions satisfying the following Equations? $$\frac{\partial v}{\partial x^l}+ v\frac{\partial u}{\partial y^l} +u\frac{\partial v}{\partial y^l}=0,\hspace{1cm} \forall l, \hspace{1cm} 1‎\leq l‎\leq n$$ and $$\frac{\partial u}{\partial x^l}- v\frac{\partial v}{\partial y^l} +u\frac{\partial u}{\partial y^l}=0,\hspace{1cm} \forall l, \hspace{1cm} 1‎\leq l‎\leq n$$