Zariski-van Kampen Method and Monodromy in Complexified Integrable Systems

We computed the fundamental groups of non-singular sets of some complexified Hamiltonian integrable systems by Zariski-van Kampen method. By our computation, we determined all possible monodromy in complexified planar Kepler problem and spherical pendulum, that is their monodromy groups. We also gave an answer to the conjecture proposed by You and Sun.