Open Problems in Integrable Systems — Nijenhuis operators: singular points and global properties

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.

For $\sigma=(\sigma_1,\ldots,\sigma_n)$, define the companion matrix $C(\sigma)$ by \[ C_{i1}=-\sigma_i\quad(1\leq i\leq n),\qquad C_{i,i+1}=1\quad(1\leq i<n), \] with all remaining entries zero. For a map $\Phi=(\sigma_1,\ldots,\sigma_n)$ in coordinates $x_1,\ldots,x_n$, its Jacobian is $J_{ij}=\partial\sigma_i/\partial x_j$.

Let $U$ be a neighborhood of $0$ in $\mathbb R^n$ and $\Phi:U\to\mathbb R^n$ be smooth, with $\operatorname{rank}J(0)<n$ and $\det J\neq0$ almost everywhere on $U$. On the regular set define \[ L(x)=J(x)^{-1}C(\sigma(x))J(x). \] This is the reconstruction of the Nijenhuis field from characteristic coefficients in equation (5.5).

Statement recorded in the supplied source.

Find necessary and sufficient conditions on $\Phi$ for every entry of $L$, initially defined where $\det J\neq0$, to extend to a smooth function on all of $U$. Describe all such maps, including their behavior at the singular point $0$. [S2]

Short English statement

Which generically regular coefficient maps yield a Nijenhuis tensor whose companion-matrix reconstruction stays smooth across all singular points?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2100512 (AMR-020-0512). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Alexey Bolsinov, Vladimir S. Matveev, Eva Miranda and Serge Tabachnikov, Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems (2018), Problem 5.14 and its complete preceding definitions. Original question and full discussion. https://arxiv.org/abs/1804.03737.