Eleven different types of ''maximally superintegrable'' Hamiltonian sy
stems on the real hyperboloid (s(0))(2) - (s(1))(2) + (s(2))(2) - (s(3
))(2) = 1 are obtained. All of them correspond to a free Hamiltonian s
ystem 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 wh
ich it is possible to form two or more triplets in involution (each of
them includes the hamiltonian). The corresponding classical and quant
um equations of motion can be solved by separation of variables on the
O(2, 2) space.