Nijenhuis Geometry and Integrable Systems — Problem 6

Definitions and mathematical statement

The supplied record asks about the explicit normal forms in a complete classification of two-dimensional $\mathrm{gl}$-regular Nijenhuis operators. Neither the normal forms nor the definitions of conservation law and regular conservation law are included in the record. The following is a conditional local formulation using standard differential-geometric terminology.

Let $M$ be a smooth real manifold of dimension $2$, and let $L\colon TM\to TM$ be a smooth field of endomorphisms. Its Nijenhuis torsion is \[ N_L(X,Y)=[LX,LY]-L[LX,Y]-L[X,LY]+L^2[X,Y] \] for smooth vector fields $X,Y$. The operator $L$ is Nijenhuis if $N_L=0$. It is pointwise $\mathrm{gl}$-regular if, at every point $p$, the minimal polynomial of $L_p$ equals its characteristic polynomial and therefore has degree $2$.

For a one-form $\alpha$, define $(L^*\alpha)(X)=\alpha(LX)$. Under the interpretation that a scalar conservation law is a function $f$ satisfying \[ {\,\mathrm{d}}(L^*{\,\mathrm{d}} f)=0, \] the requested classification is: for every normal form $L$ in the classification referenced by the source, determine all local functions $f$ satisfying this equation. More explicitly, on a neighborhood $U$ where that normal form is defined, determine \[ \mathcal C_L(U)=\{f\in C^\infty(U,{\mathbb{R}}):{\,\mathrm{d}}(L^*{\,\mathrm{d}} f)=0\}. \] Equivalently in this local interpretation, one may seek closed one-forms $\alpha$ for which $L^*\alpha$ is also closed; such forms are locally of the form $\alpha={\,\mathrm{d}} f$.

If a regular conservation law means one whose differential is a cyclic covector for $L^*$, the weaker requested task is: for each such normal form and each point $p$ under consideration, find a neighborhood $U$ of $p$ and $f\in\mathcal C_L(U)$ such that \[ ({\,\mathrm{d}} f\wedge L^*{\,\mathrm{d}} f)_p\ne0. \] After shrinking $U$, this condition holds throughout $U$.

The smooth real category, the local interpretation, the cyclic-covector meaning of regularity, and the requirement at every point are not specified in the supplied record. They are explicit conditional interpretations, not established source hypotheses. The intended classification and its admissible domains must be supplied before these tasks can be instantiated for each normal form.

Short English statement

For each explicit two-dimensional $\mathrm{gl}$-regular Nijenhuis normal form referenced by the source, describe all conservation laws, or at least find one regular conservation law. The precise normal forms, function category, locality, and meaning of regularity are missing; the displayed differential equations give a conditional local interpretation.

Sources

• Local UnsolvedMath registry: permanent ID $801153$, source problem code catalogue entry, title "Nijenhuis Geometry and Integrable Systems --- Problem 6." The supplied record attributes the problem to Bolsinov, Konyaev, and Matveev, with proposed year $2024$.
• Bolsinov, Konyaev, and Matveev, Research Problems on Nijenhuis Geometry and Integrable Systems (2024), Problem 6. Supplied source link: https://arxiv.org/abs/2410.04276. The background records access on 2026-10-16 and extraction method arxiv_latex; these are supplied provenance claims, not independently verified access or source checking.
• The registry labels the problem "open" and records upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82, with record update dated 2026-10-16. This status is supplied curation, not an independently verified literature assessment. The statement's assertion that a complete classification is available is likewise a supplied source assertion.