I need some help with the theory of PDE systems, which I am not very familiar with.
Assume $f$ and $g$ are two functions of $x,y$ in the real plane. I have the equation
$$\nabla\cdot\left(\frac{1}{\sqrt{1-g^2}}\nabla g\right)=0$$
Now, this clearly looks singular because when $g=1$ I get in trouble. An alternative I have is to use that $f=\sqrt{1-g^2}$ so that the following system holds
$$\nabla\cdot\left(\frac{1}{f}\nabla g\right)=0$$ $$\nabla\cdot\left(\frac{1}{g}\nabla f\right)=0$$
and I know that whenever $g=1, f\neq 1$ and viceversa. In this case the matrix of the system seems non-singular and the issue looks resolved.
Another solution I had in mind was to match the points where the first equation (in terms of $f$ only) becomes singular by using the second equation (in terms of $g$ only), and then put them together using $f=\sqrt{1-g^2}$. This sounds trickier to me, and I do not know if the whole matching story is that mathematically neat.
Is, from a rigorous point of view the system still singular. And in that case, what condition should I impose to avoid the singularity? Or have I solved the singularity issue by introducing $g$? Thanks.