Nonintegrability of the classical Zeeman Hamiltonian

We prove that the Hamiltonian H of the three dimensional hydrogen atom in a uniform static magnetic field B does not have an integral which (i) is real analytic on the phase space M of the system; (ii) is in involution with the component M3 of the angular momentum along B; (iii) is functionally independent of H and M3 and (iv) has a meromorphic (single-valued) extension to the complexification of M in bfC6. This follows from the fact that the Hamiltonian KM of two degrees of freedom obtained by fixing M3 at certain nonzero values M and reducing H w.r. to the rotational symmetry about the magnetic field, has a complexification which is nonintegrable in the Ziglin sense. We prove this nonintegrability by demonstrating that for each such M the monodromy group of the normal variational equation along a certain complexified phase curve of KM is not Ziglin, using Churchill and Rod's adaptation of Kovacic's algorithm to the Ziglin analysis. Analogous arguments prove that the Hamiltonian of the Stoermer problem is nonintegrable in the same sense. (orig.)