I have a hamiltonian of the form $$H(\phi,z) = (1-z^2)\cos(2\phi) + \chi z^2$$ with position $\phi$ and conjugate momentum $z$. It has this form provided that $z \in [-1,1]$ and we have natural representation of the dynamics on sphere with $z$ being distance on the $z$-axis and $\phi$ - azimuthal angle.
Now, I am looking for equilibrium points (fixed points) and one of them includes $z = 1$ or $z=-1$ and some $\phi$. I am not sure if I can call such points fixed points cause phase $\phi$ is not defined on poles. How to find the approximate dynamics around poles (it depends on $\chi$) ?