Consider the non linear, 1st order PDE system
\begin{align} \xi_u^2+\eta_u^2=\xi_v^2+\eta_v^2=\left(1+\frac{\xi^2+\eta^2}{4} \right)^2, \end{align}
for $\xi=\xi(u,v)$ and $\eta=\eta(u,v)$, with $u,v \in \mathbb{R}^2$. What type of system is it ? Most common PDE classification rules apply to second order equations, linear or semi-linear, which is not the case here. Alternatively, what are the conditions for existence and uniqueness of the solutions on some (compact, although not necessarily) $(u,v)$ domain ?