Open Problems in Integrable Systems — Around the Birkhoff conjecture

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $\gamma\subset\mathbb{RP}^2$ be a smooth closed strictly convex curve in an affine chart, with a smooth exterior foliation $\mathcal F$ by closed convex curves including $\gamma$ as a leaf. For each tangent line $L$ to $\gamma$, pairing the two nearby intersections of $L$ with the same leaf defines a local involution, fixing the tangency point. Assume this involution is the restriction of a projective transformation of $L$. A first integral is a nonconstant function whose level curves are the foliation leaves. A rational integral is rational in affine coordinates; an algebraic integral is a local branch of a function satisfying a nonzero polynomial equation over the rational-function field. A pencil of conics consists of equations $aQ_0+bQ_1=0$ for fixed homogeneous quadratic forms $Q_0,Q_1$.

Statement recorded in the supplied source.

Prove that $\gamma$ is a conic (an ellipse in the affine chart) and that $\mathcal F$ consists of conics in one pencil when the foliation admits (i) a rational first integral or (ii) an algebraic first integral. These are the two specified special cases of the source's projective-involution conjecture. [S2]

Short English statement

Prove the conic-pencil conclusion for a convex exterior foliation with projective tangent-line involutions and a rational or algebraic first integral.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2100403 (AMR-020-0403). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 A. Bolsinov, V. S. Matveev, E. Miranda and S. Tabachnikov, Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems (2018), arXiv:1804.03737, Problem 4.5 and Conjecture 4.3. https://arxiv.org/abs/1804.03737.