Open Problems in Integrable Systems — Argument shift method and Jordan-Kronecker invariants of Lie algebras

Definitions and mathematical statement

In a finite-dimensional polynomial setting with coordinates $x_1,\ldots,x_d$, a Poisson bivector $P$ defines $\{f,g\}_P=P({\,\mathrm{d}} f,{\,\mathrm{d}} g)$, satisfying the Jacobi identity. Two such bivectors $P,Q$ are compatible if every linear combination is Poisson; over a characteristic-zero field this can be expressed by \[ [P,P]=[Q,Q]=[P,Q]=0 \] using the Schouten bracket. Their Poisson pencil is $P_\lambda=P+\lambda Q$; the span of $P,Q$ also includes the parameter value at infinity.

The source asks to study examples of "quadratic $+$ linear" pencils, compute their algebraic types, and construct complete polynomial families in bi-involution. In the homogeneous-coordinate interpretation, the coefficients of $P$ are homogeneous quadratic polynomials and those of $Q$ are homogeneous linear polynomials. The record does not specify whether this homogeneity is required or whether quadratic and linear instead mean degree at most two and one, nor does it give a ground field, a dimension, or a particular pencil.

At a point $x$, the pair $P(x),Q(x)$ is a pair of skew forms on the cotangent space. In the Jordan--Kronecker interpretation suggested by the source title, algebraic type records the canonical block data of this pair, including the relevant Jordan and Kronecker data. A precise computation must specify the allowed equivalence and whether the target is the generic type, special-point types, or the whole stratification.

A polynomial family $f_1,\ldots,f_s$ is in bi-involution if \[ \{f_i,f_j\}_P=\{f_i,f_j\}_Q=0\qquad(1\le i,j\le s), \] so it commutes for every member of the pencil. Functional independence is measured by the span of ${\,\mathrm{d}} f_i(x)$ at generic points. One usual completeness target, if the generic pencil rank is $r$, is a common isotropic differential span of dimension $d-r/2$; an alternative formulation uses Liouville completeness on specified generic symplectic leaves together with the relevant Casimirs. The source does not state which completeness convention, exceptional parameter values, or rank assumptions are intended, so none is silently fixed as a universal requirement.

This is an example-building and classification program: specify interesting compatible quadratic--linear pairs, determine their types, and seek complete polynomial bi-involutive families for the chosen examples. The extract does not assert that every such pair admits a complete polynomial family or request a single fully quantified classification theorem. The supplied summary reports no verified systematic resolution, while citing related quadratic-bracket frameworks as context; this is a source-verification limitation, not a proof of openness.

Short English statement

Explore Poisson pencils formed from a quadratic and a linear bracket, compute their Jordan--Kronecker data, and find enough independent polynomials commuting for both brackets. The source gives no explicit pencil, field, dimension, degree convention, or precise completeness requirement, so the task remains a program of specified examples rather than a universal integrability theorem.

Sources

• Local source registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $2100506$, source code catalogue entry, including the supplied OPEN-TRIAGE summary. Dataset attribution: Ulam AI and UnsolvedMath Contributors; original-source attribution and rights are retained.
• Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems (2018), Problem 5.6: https://arxiv.org/abs/1804.03737. The metadata reports source-TeX extraction and July 29, 2026 access, with status and rights review flags; no author list is supplied in this record.
• The supplied summary mentions Sklyanin brackets, quadratic $r$-matrix brackets, Poisson--Lie groups, and comparison with the purely linear Lie--Poisson case. It supplies no separate precise references or verified theorem settling the stated quadratic--linear program. No linked source was inspected for this draft.