  The three-dimensional Kepler problem is related to the four-dimensional
isotropic harmonic oscillators by the Kustaanheimo-Stiefel Transformations. In
the first part of this paper, we study how certain integrable mechanical
billiards are related by this transformation. This in part illustrates the
rotation-invariance of integrable reflection walls in the three-dimensional
Kepler billiards found till so far. The second part of this paper deals with
Birkhoff-Waldvogel's Transformation of the three-dimensional problem with two
Kepler centers. In particular, we establish an analogous theory of Levi-Civita
planes for Birkhoff-Waldvogel's Transformation and showed the integrability of
certain three-dimensional 2-center billiards via a different approach.
