Open Problems and Questions about Geodesics — Conjecture 61

Definitions and mathematical statement

Let $\Omega$ denote the billiard domain in the supplied source. The excerpt does not specify its ambient space, dimension, boundedness, convexity, or boundary regularity. Let $X$ denote the intended phase space, described in the excerpt as the tangent bundle, and write $\Phi^t$ for billiard evolution whenever it is defined. A smooth first integral is a function $F\colon X\to{\mathbb{R}}$ satisfying \[ F(\Phi^t(z))=F(z) \qquad\text{for every }z\in X\text{ and every admissible time }t. \] The supplied excerpt describes integrability by the existence of such an $F$ that is nonconstant.

The source's proposed implication is \[ \text{the billiard in }\Omega\text{ is integrable} \quad\Longrightarrow\quad \partial\Omega\text{ is an ellipse}. \] In the planar Euclidean interpretation suggested by the conclusion, an ellipse means a set \[ \left\{c+Q(a\cos\theta,b\sin\theta):0\leq\theta<2\pi\right\}, \qquad c\in{\mathbb{R}}^2,\quad a,b>0,\quad Q^{\mathsf T}Q=I. \] This interpretation of the ambient geometry is not explicitly specified in the supplied excerpt.

Unresolved qualifications. The displayed implication is a transcription of the proposed target, not a complete formulation with established hypotheses. For the usual Euclidean billiard with elastic reflection, squared speed is already a nonconstant smooth first integral on the full tangent bundle. Thus the excerpt's nonconstancy requirement does not identify the intended nontrivial notion of integrability. It remains unspecified whether the phase space should have fixed speed, whether the integral must be functionally independent of energy, or whether another condition is intended. The treatment of reflection points and singular trajectories, and the meaning of smoothness there, are also unspecified. These missing qualifications must be resolved before the conjecture has a precise intended meaning.

Short English statement

The supplied conjecture says that an integrable billiard must have an elliptical boundary, and gives elliptical billiards as known integrable examples. The excerpt omits the domain hypotheses and a sufficiently specified notion of integrability: on the full Euclidean tangent bundle, speed already provides a nonconstant conserved quantity. The intended conjecture therefore requires clarification.

Sources

• Local UnsolvedMath registry, permanent ID $800980$, source problem code catalogue entry, upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82. The supplied record attributes the problem to Burns and Matveev, gives proposed year 2019, and labels it open; this status is supplied metadata, not an independently verified literature assessment.
• Burns and Matveev, Open Problems and Questions about Geodesics (2019, as identified in the supplied record), Conjecture 61. Supplied source URL: https://arxiv.org/abs/1308.5417. The record reports access on 2026-10-16 and extraction by arxiv_latex; these are supplied provenance statements, not a claim of independent source access.
• The excerpt cites the reference label Tabachnikov2005 for the integrability of billiards in ellipses. No full bibliographic entry or corresponding URL is supplied, so this reference cannot be identified precisely from the record alone.