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Strong-field analogue of the magnetic Hopf-type rigidity

Establish an analogue of Theorem \ref{main} for magnetic billiards in strong constant magnetic fields, proving that total integrability forces the boundary to be a circle.

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Background

Theorem \ref{main} shows that for a weak constant magnetic field, total integrability of the magnetic billiard implies the boundary is a circle.

The authors inquire whether an analogous rigidity holds for strong constant magnetic fields, noting that the algebraic approach suggests an affirmative answer.

References

In this section we formulate natural open questions related to the results discussed in previous sections. (1) Is it possible to state an analog of Theorem \ref{main} for strong constant magnetic fields? The algebraic approach hints that this is indeed the case.

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