At high energies, every real-analytic convex Kepler billiard is non-integrable for all but at most one position of the center unless the table is an ellipse with the center at a focus.
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On the Birkhoff conjecture for Kepler billiards
At high energies, every real-analytic convex Kepler billiard is non-integrable for all but at most one position of the center unless the table is an ellipse with the center at a focus.