  For a class of $O(n+1,R)$ invariant measures on the Kepler manifold
possessing finite moments of all orders, we describe the reproducing kernels of
the associated Bergman spaces, discuss the corresponding asymptotic expansions
of Tian-Yau-Zelditch, and study the relevant Hankel operators with conjugate
holomorphic symbols. Related reproducing kernels on the minimal ball are also
discussed. Finally, we observe that the Kepler manifold either does not admit
balanced metrics, or such metrics are not unique.
