Suppose, for example, that I have the following non-linear system of ODE's;
$$(y^2-y'^2)g+y' y g=P$$ $$y'(y' g''+(y''+y)g')=S,$$
where $y,g,P,S$ are all functions of the dependent variable $x$. How can I find the possible singular points of this system, i.e. points where $y$ and $g$ might diverge?
For example, in the linear equation
$$y''+a(x) y' +b(x)y=0$$
these points would be the poles of $a(x)$ and $b(x)$. What's the case for non-linear equations?