The analysis reveals that Kepler billiards are integrable at zero-energy, suggesting they exist within elliptical shapes with specific focal points.
We show that if a Kepler billiard in the plane defined inside a smooth closed connected simple curve intersecting all focused parabola at at most two points is integrable at its zero-energy, then the system is defined actually inside an ellipse with the Kepler center occupying one of the foci. This statement is obtained as a simple ``translation'' of the theorem of Bialy-Mironov 2022 with Levi-Civita transformation.
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Lei Zhao (2025) studied this question.
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