Integrability of Kepler Billiards at Zero-Energy

Abstract: 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.