(This question comes as a particular case with specific boundary conditions of the system shown in mathSE)

Consider the PDE system

\begin{align} \xi_u^2+\eta_u^2&=\left(1+\frac{\xi^2+\eta^2}{4} \right)^2 \\ \xi_v^2+\eta_v^2&=\left(1+\frac{\xi^2+\eta^2}{4} \right)^2\\ \xi(0,v)=\eta(u,0)&=0, \end{align}

for $\xi=\xi(u,v)$ and $\eta=\eta(u,v)$, with $u,v \in \mathbb{R}^+$. From the form of the boundary condition and some numerical evidence, we can propose the Ansatz $\eta(u,v)=\xi(v,u)$ and try to get a single 1st order PDE as

\begin{align} \xi_u^2(u,v)+\xi_u^2(v,u)&=\left(1+\frac{\xi^2(u,v)+\xi^2(v,u)}{4} \right)^2. \end{align}

Does it make sense to tackle a PDE like this one where the function appears with the arguments swapped? My goal is to find an additional convenient Ansatz that may reduce it to some ODE, even if it may not have an analytical solution.

(This question comes as a particular case with specific boundary conditions of the system shown in mathSE)

Consider the PDE system $$ \begin{cases} \xi_u^2+\eta_u^2=\left(1+\dfrac{\xi^2+\eta^2}{4} \right)^2 \\ \xi_v^2+\eta_v^2=\left(1+\dfrac{\xi^2+\eta^2}{4} \right)^2\\ \\ \xi(0,v)=\eta(u,0)=0, \end{cases} $$ for $\xi=\xi(u,v)$ and $\eta=\eta(u,v)$, with $u,v \in \mathbb{R}^+$. From the form of the boundary condition and some numerical evidence, we can propose the Ansatz $\eta(u,v)=\xi(v,u)$ and try to get a single 1st order PDE as

\begin{align} \xi_u^2(u,v)+\xi_u^2(v,u)&=\left(1+\frac{\xi^2(u,v)+\xi^2(v,u)}{4} \right)^2. \end{align}

Does it make sense to tackle a PDE like this one where the function appears with the arguments swapped? My goal is to find an additional convenient Ansatz that may reduce it to some ODE, even if it may not have an analytical solution.