I came across the following statement in Olver's "Applications of Lie Groups to Differential Equations". Paraphrasing:
Let $\xi^1(x,y)=x+y$, $\xi^2(x,y)=\frac{x}{y}$, and $U$ an open subset of $\mathbb{R}^2$ which does not contain the line $y\ne 0$. For any smooth (infinitely differentiable) $F:\mathbb{R}^2\rightarrow\mathbb{R}$, if $F\left(\xi^1(x,y),\xi^2(x,y)\right)=0$ for $(x,y)\in U$, then there exists an open subset of $V=\left(\xi^1(U),\xi^2(U)\right)$ such that $F(V)=0$.
Apparently this follows through an application of the inverse function theorem, but I can't see how to make this work.