Action-angle coordinates of spherical pendulums with symmetric quadratic potentials

We study the spherical pendulum system with an arbitrary potential function $V=V(z)$, which is an integrable system with a first integral whose Hamiltonian flow is periodic. We give an explicit solution to this integrable system and then we compute its action-angle coordinates. In the special case where the potential function is symmetric quadratic like $V=z^2$, we represent its action-angle coordinates in terms of elliptic integrals, and calculate the monodromy.