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

Definitions and mathematical statement

The supplied Problem 5.11 asks to classify functions $f(x,y)$ near $(0,0)\in{\mathbb{R}}^2$ satisfying the three displayed conditions \[ f_y(0,0)\not\equiv0,\qquad f_y(0,0)=0,\qquad \frac{f_x(x-f_x)-f}{f_y}\text{ extends to a locally smooth function}. \] The first two conditions are incompatible if both refer to the scalar value $f_y(0,0)$: a scalar nonzero condition and equality to zero cannot both hold. The symbol $\not\equiv$ normally applies to a function, so a plausible intended first condition is that $f_y$ is not identically zero as a local function, while its value at the origin is zero. That repair is not established by the supplied record and is not silently substituted for the literal text.

For precision about the third condition, under the product interpretation of the numerator write \[ N_f(x,y)=f_x(x,y)\left(x-f_x(x,y)\right)-f(x,y), \qquad U_f=\{(x,y)\in U:f_y(x,y)\ne0\}, \] where $U$ is a neighborhood of the origin on which the required derivatives exist. A smooth-extension condition for the quotient means that, after possibly shrinking $U$, there is $g\in C^\infty(U)$ such that \[ g(x,y)=\frac{N_f(x,y)}{f_y(x,y)} \quad\text{for all }(x,y)\in U_f. \] It does not automatically require $N_f=f_yg$ at points outside the closure of $U_f$. In particular, the hypothesis that $f_y$ is not identically zero does not by itself make $U_f$ dense. A global smooth-divisibility formulation would require additional information and cannot simply replace the extension wording.

The source does not specify the regularity class of $f$ itself or whether "locally smooth" means a smooth germ at the origin. It also does not explicitly expand $f_x(x-f_x)$; the product interpretation above is natural, whereas reading it as evaluation would need additional arguments. The supplied summary relates the condition to Nijenhuis/recursion operators at singularities and reports no verified published classification. It supplies no operator formula, torsion condition, or precise reference, so none is added to the question.

Short English statement

The source asks which functions make a derivative quotient smoothly extend across zeros of $f_y$. Its first two displayed conditions conflict as written; a likely nonidentically-zero derivative condition needs verification. The extension requirement is stated exactly on the quotient's domain, without assuming an unprovided divisibility condition on an open zero set.

Sources

• Local UnsolvedMath registry, permanent ID 2100511, exact source code catalogue entry, "Open Problems in Integrable Systems --- Nijenhuis operators: singular points and global properties"; status open, classification OPEN-TRIAGE.
• Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems (2018), Problem 5.11, https://arxiv.org/abs/1804.03737. The record identifies extraction from public arXiv source TeX, accessed July 29, 2026, and says the source presented this as open.
• The supplied research summary reports the classification open/unverified and connects it with a Nijenhuis operator associated with a map $\Phi$, Nijenhuis torsion, and recursion-operator smoothness at singular points. It supplies neither that map nor a precise theorem or citation, and says it could not verify a published classification.