I'm studying the way one changes a general 2nd-order, 2-dimensional, linear hyperbolic PDE $$a(x,y)u_{xx} + 2b(x,y) u_{xy} + c(x,y) u_{yy} + LOT = 0$$ into the canonical form $$A(\xi,\eta)u_{\xi\xi} + 2B(\xi,\eta)u_{\xi\eta} + C(\xi,\eta)u_{\eta\eta} + LOT = 0$$ through the change of variables $(x,y)\mapsto(\xi(x,y),\eta(x,y))$. Here LOT means lower order terms.
For hyperbolic PDE, we assume $b^2-ac>0$ everywhere in the relevant domain. Let's also assume $a\neq0$. I understand that for the change to canonical form to occur, $\xi$ and $\eta$ must satisfy the PDEs \begin{align}\tag{$*$} \xi_x - \lambda_1\xi_y = 0 \\ \eta_x - \lambda_2\eta_y = 0 \end{align} where $$\lambda_{1,2} = \frac{-b\pm\sqrt{b^2-ac}}{a}.$$
QUESTION: It seems that all sources I read assume that a valid change of coordinates exists for every such hyperbolic PDE. But how can we guarantee that there exists a solution to $(*)$ which is truly a change of coordinates, i.e., invertible (with $\xi_x\eta_y-\xi_y\eta_x\neq0$ everywhere)?
If we know $\xi_y,\eta_y\neq0$, then we have $$\xi_x\eta_y-\xi_y\eta_x = \xi_y\eta_y(\lambda_1-\lambda_2) = \frac{2\xi_y\eta_y\sqrt{b^2-ac}}{a} \neq 0$$
I know that as a first-order equation, $(*)$ has unique (local) solutions given non-characteristic initial value data. Is there a way to tune that initial data to ensure that $\xi_y,\eta_y\neq0$ everywhere?
Thanks for the help.