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Beyond ellipses: totally integrable outer length billiards

Determine whether totally integrable outer length billiards exist other than ellipses.

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Background

For the outer length billiard (fourth billiard), ellipses are known to be totally integrable with invariant confocal foliations.

The authors ask whether there are additional shapes whose outer length billiard is totally integrable.

References

In this section we formulate natural open questions related to the results discussed in previous sections. (1) For outer length billiards, ellipses are totally integrable. Are there other totally integrable billiards?

— Integrable Billiards and Related Topics (2510.03790 - Bialy et al., 4 Oct 2025) in Section 9 (Open questions), Subsection Outer length billiards, item (1)