  We study the regularized $(2n-1)$-Kepler problem and other Hamiltonian
systems which are related to the nilpotent coadjoint orbits of $U(n,n)$. The
Kustaanheimo-Stiefel and Cayley regularization procedures are discussed and
their equivalence is shown. Some integrable generalization (perturbation) of
$(2n-1)$-Kepler problem is proposed.
