Let $(\mathbb{R}^2,\langle .,.\rangle)$ be the Euclidean space and define the almost complex structure $J_{\delta,\beta}:TT\mathbb{R}^2\longrightarrow TT\mathbb{R}^2$ with \begin{align} J_{\delta,\beta}(X^h)=\beta X^h +\alpha X^v\\ J_{\delta,\beta}(X^v)=-\beta X^v -\delta X^h, \end{align} where $X^h,X^v$ are the horizontal and vertical lifts of the vector $X\in T\mathbb{R}^2$ and $\alpha , \delta, \beta : T\mathbb{R}^2 \longrightarrow \mathbb{R}$ are functions satisfy in $\alpha \delta - \beta ^2=1$. Then $J_{\delta, \beta}$ is integrable if and only if $\delta , \beta$ satisfy in the expressed PDE when $d\beta \neq 0$.

Let $δ(x^1,x^2,y^1,y^2)$ and $β(x^1,x^2,y^1,y^2)$ be two functions. Are there $δ$ and $β$ which satisfy in the following PDE system?

\begin{align} \frac{\partial \beta}{\partial y^i}-\frac{\beta}{\delta}\frac{\partial \delta}{\partial y^i}=0 \hspace{1cm}i=1,2 \end{align}

\begin{align} \frac{\partial \delta}{\partial y^i}=-\delta ^2 \hspace{1cm}i=1,2 \end{align} \begin{align} \frac{\partial \beta}{\partial x^i}-\frac{\beta}{\delta}\frac{\partial \delta}{\partial x^i}=1 \hspace{1cm}i=1,2 \end{align} \begin{align} \frac{\partial \delta}{\partial x^i}=\beta \delta \hspace{1cm}i=1,2 \end{align}