Open Problems in Integrable Systems — Polynomial in momenta integrals in higher dimensions.

Definitions and mathematical statement

Let $(M,g)$ be a connected symmetric space in the intended metric category, and let $\nabla$ be its Levi--Civita connection. A symmetric contravariant tensor $K$ of rank $r\ge1$ is a Killing tensor when \[ \nabla^{(i_0}K^{i_1\cdots i_r)}=0. \] A Killing vector field $X$ satisfies $\mathcal L_Xg=0$. Write $\odot$ for the symmetrized tensor product. The question is whether, for every such $M$, rank $r$, and Killing tensor $K$, there exist finitely many Killing vector fields and real constants such that \[ K=\sum_{j=1}^N a_j\, X_{j,1}\odot\cdots\odot X_{j,r},\qquad a_j\in{\mathbb{R}}. \] The coefficients are constants, not arbitrary functions on $M$.

Equivalently, on $T^*M$ with its canonical Poisson bracket and geodesic Hamiltonian \[ H(x,p)=\tfrac12 g^{ij}(x)p_ip_j, \] let $\mathcal I_{\mathrm{poly}}$ be the real algebra of smooth fiberwise-polynomial functions of finite degree satisfying $\{H,I\}=0$. For a Killing vector $X$, the corresponding linear integral is $I_X(x,p)=p(X_x)$. The equivalent algebra-generation question is \[ \mathcal I_{\mathrm{poly}}\stackrel{?}{=}{\mathbb{R}}[I_X:X\text{ a Killing vector field}], \] with constants included. Homogeneous polynomial integrals correspond to Killing tensors by $I_K(x,p)=K^{i_1\cdots i_r}(x)p_{i_1}\cdots p_{i_r}$.

The supplied research summary reports a negative answer in general, attributing counterexamples to Matveev--Nikolayevsky, arXiv:2312.16518. It supplies no particular counterexample or its hypotheses. The excerpt does not specify Riemannian versus pseudo-Riemannian signature, local versus global symmetric spaces, or local versus global tensors; the global-tensor reading above requires the corresponding source category to be checked.

Short English statement

Are all Killing tensors on a symmetric space built from Killing vector fields using sums and symmetric products? Equivalently, are polynomial geodesic integrals generated by linear ones? The supplied summary reports counterexamples to the general assertion.

Sources

• Local UnsolvedMath registry, permanent ID $2100306$, exact source code catalogue entry, "Open Problems in Integrable Systems --- Polynomial in momenta integrals in higher dimensions."
• Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems (2018), Question 3.6; https://arxiv.org/abs/1804.03737. Source-TeX extraction; recorded access 2026-07-29.
• Supplied research summary: Matveev--Nikolayevsky, arXiv:2312.16518, reported counterexamples. No title, precise example, or theorem number is supplied.