Let R-*(2k+1) = R2k+1 backslash {(0) over right arrow} (k >= 1) and pi: R-*(2k+1) -> S-2k be the map sending (r) over right arrow is an element of R-*(2k+1) to (r) over right arrow/(r) over right arrow is an element of S-2k. Denote by P -> R-*(2k+1) the pullback by pi of the canonical principal SO(2k)-bundle SO(2k + 1) -> S-2k. Let E-# -> R-*(2k+1) be the associated co-adjoint bundle and E-# -> T*R2k+1 be the pullback bundle under projection map T*R-*(2k+1) -> R-*(2k+1). The canonical connection on SO(2k + 1) -> S-2k turns E-# into a Poisson manifold. The main result here is that the real Lie algebra so(2, 2k + 2) can be realized as a Lie subalgebra of the Poisson algebra (C-infinity(O-#), {,}), where O-# is a symplectic leave of E-# of special kind. Consequently, in view of the earlier result of the author, an extension of the classical MICZ Kepler problems to dimension 2k + 1 is obtained. The Hamiltonian, the angular momentum, the Lenz vector, and the equation of motion for this extension are all explicitly worked out. (C) 2013 AIP Publishing LLC.
