For a function such as $f(z)=z^{1/2}$ it is easy and intuitive to me how to find its branch points. Now, I have to deal with a complex curve with the defining equation $$ z + \frac{1}{z} = \frac{y^2-u}{c^2} $$ where $c$ is a constant other than zero and $z \in \mathbb{P}^1$. I would like to consider a function $y(z)$ and find its branch points. I can immediately see that there are two points that blow the curve up, that is $0$ and $\infty$. Then there must be another two points from what I read and they are found in the approximation that $|u| \gg c^2$.
How can I find these extra two branch points and why we need to consider this approximation?