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New integrable examples of outer billiards

Construct integrable outer billiard tables in the plane other than ellipses, or prove that none exist.

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Background

Ellipses are known to yield integrable outer billiards (via affine invariance). The algebraic version of the integrability question has a negative answer: only ellipses are algebraically integrable.

The authors ask whether genuinely new (non-elliptic) integrable outer billiards exist in the smooth setting.

References

In this section we formulate natural open questions related to the results discussed in previous sections. (1) Are there new integrable examples of outer billiards?

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