Abstract
We construct two new examples of integrable Hamiltonian systems. They describe the motion of a particle under the influence of certain quartic potentials. The first system describes such a motion on the n-dimensional sphere. The second describes the motion on the n-dimensional hyperbolic space. Here n is an arbitrary positive integer. We represent the system on a (2n + 1)-dimensional sphere with an additional U(1)-symmetry as a symplectic reconstruction of a system which has a topologically non-trivial magnetic term and whose configuration space is the n-dimensional complex projective space. We use this description to give an alternative proof of the integrability of the system on the odd-dimensional sphere.
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