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

Kustaanheimo-Stiefel Transformation, Birkhoff-Waldvogel Transformation and Integrable Mechanical Billiards