Intersection of curve $x'=0$ with orbits of a dynamical system

Question

Let us suppose I have the following dynamical system $$ x'=f(x,y)\\ y'=g(x,y) $$

The curve $x'=0$ which is not an orbit of the system will intersect orbits at critical points (where $x'=0$ and $y'=0$). Is there a general way to analyse where else the curve $x'=0$ will intersect the orbits of the system