The Conformal Group SU(2, 2) and Integrable Systems on a Lorentzian Hyperboloid

Eleven different types of “maximally superintegrable” Hamiltonian systems on the real hyperboloid (s⁰)² – (s¹)² + (s²)² – (s³)² = 1 are obtained. All of them correspond to a free Hamiltonian system on the homogeneous space SU(2, 2)/U(2, 1), but to reductions by different maximal abelian subgroups of SU(2, 2). Each of the obtained systems allows 5 functionally independent integrals of motion, from which it is possible to form two or more triplets in involution (each of them includes the hamiltonian). The corresponding classical and quantum equations of motion can be solved by separation of variables on the O(2, 2) space.