Hamilton's equations are
$\begin{align*}\dot{\theta} &= \frac{\partial H}{\partial p_{\theta}} \\\dot{p}_{\theta} &= -\frac{\partial H}{\partia...$
$H$ does not depend on $\phi$ , so the last equation implies that
$\begin{align*}\dot{p}_{\phi} = 0\end{align*}$
or
$\begin{align*}p_{\phi} = \text{const}\end{align*}$
Therefore, answer (E) is correct.

Solution to 2008 Problem 83