Spherical phase space dynamics 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 $\chi$) ?
 A: Hint :
\begin{equation}
H(\phi,z) = (1-z^2)\cos2\phi + \chi z^2
\tag{01}
\end{equation}
\begin{equation}
\left.
\begin{cases}
\overset{\boldsymbol{.}}{z}=\boldsymbol{-}\dfrac{\partial H}{\partial \phi}=\boldsymbol{+}2\left(1-z^{2}\right)\sin2\phi \vphantom{\dfrac{\dfrac{a}{b}}{b}}\\
\overset{\boldsymbol{.}}{\phi}=\boldsymbol{+}\dfrac{\partial H}{\partial z}=\boldsymbol{-}2\left(\cos2\phi-\chi\right)z\vphantom{\dfrac{\dfrac{a}{b}}{b}}
\end{cases}
\right\}
\tag{02}
\end{equation}
To find fixed points the two equations in (03) must be fulfilled simultaneously 
\begin{equation}
\left.
\begin{cases}
\overset{\boldsymbol{.}}{z}=0 \vphantom{\dfrac{a}{b}}\\
\overset{\boldsymbol{.}}{\phi}=0 \vphantom{\dfrac{a}{b}}
\end{cases}\right\}
\Longrightarrow
\left.
\begin{cases}
\left(1-z^{2}\right)\sin2\phi=0  \vphantom{\dfrac{a}{b}}\\
\left(\cos2\phi-\chi\right)z=0 \vphantom{\dfrac{a}{b}}
\end{cases}\right\}
\tag{03}
\end{equation}
A: They are fixed points because the location is well defined.
You think that it is a problem only because you are represented it in $(z,\phi)$. If you write it in Cartesian coordinate, the fix points will be clear to you.
For the dynamics around a fixed point, first you should calculate the dynamic equation $(z,\dot{\phi})$ and linearize it around the point. Then you can get an approximate solution by solving two coupled linear differential equations.
