Using the Zariski-van Kampen method, the fundamental group of the regular locus is ℤ⊕ℤ for the complexified planar Kepler problem and ℤ for the complexified spherical pendulum.
The braid monodromy of plane algebraic curves and hyperplane arrange- ments
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A Remark on the Topology of the Regular Loci of Some Complexified Hamiltonian Systems
Using the Zariski-van Kampen method, the fundamental group of the regular locus is ℤ⊕ℤ for the complexified planar Kepler problem and ℤ for the complexified spherical pendulum.