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

Definitions and mathematical statement

For a pseudo-Riemannian metric $g$ with Levi-Civita connection $\nabla$, a symmetric covariant tensor $K_{a_1\ldots a_m}$ is a Killing tensor of order $m$ if \[ \nabla_{(a_0}K_{a_1\ldots a_m)}=0, \] where parentheses denote symmetrization over all displayed indices. Raising indices gives a homogeneous degree-$m$ polynomial \[ I_K(x,p)=K^{a_1\ldots a_m}(x)p_{a_1}\cdots p_{a_m} \] that Poisson-commutes with the geodesic Hamiltonian $H=\tfrac12g^{ab}p_ap_b$. An Einstein metric satisfies $\operatorname{Ric}(g)=\Lambda g$ for a constant $\Lambda$.

Construction problem. Construct stationary axially symmetric four-dimensional Einstein metrics admitting Killing tensors of higher order, interpreted as some integer $m\geq3$. In the conventional spacetime reading, stationarity is generated by a Killing field timelike in a stationary region, axial symmetry by a Killing field with periodic orbits, and the two symmetry fields commute. The source does not specify signature, the value or permitted range of $\Lambda$, local versus global existence, or global spacetime conditions; this interpretation is explicit rather than a hidden choice of those data.

A necessary clarification concerns nontriviality. Symmetrized products of lower-order Killing tensors, including Killing vector fields, already give higher-order tensors. Thus the literal rank requirement differs from asking for a genuinely new polynomial first integral. If irreducibility is intended, one must require that the proposed integral not belong to the algebra generated by the stipulated lower-order integrals, and specify that algebra. The source does not impose or define this requirement.

The supplied summary reports no verified construction or nonexistence theorem for the intended higher-order hidden symmetry. It cites the quadratic Carter symmetry of Kerr as context and qualifies its status assessment as open/unverified; it does not establish nonexistence of literal product tensors.

Short English statement

Construct four-dimensional stationary axially symmetric Einstein metrics with genuinely higher-order geodesic symmetries, if that is the intended meaning. The source must distinguish such symmetries from products of existing lower-order Killing tensors.

Sources

• Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems (2018), Question 3.4, https://arxiv.org/abs/1804.03737; source-TeX extraction, accessed 2026-07-29. No author list is supplied in this record.
• Carter (1968), cited by the research summary for the rank-two hidden symmetry of Kerr; no title or publication identifier is supplied. The summary discusses Kerr and related families, principal Killing--Yano tensors, and Kerr--NUT--(A)dS metrics without a citation resolving the higher-order problem.
• Local registry data/unsolvedmath/problems.json, permanent ID $2100304$, source code catalogue entry, Open Problems in Integrable Systems --- Polynomial in momenta integrals in higher dimensions. The record labels this open and OPEN-TRIAGE, with NEEDS_REVIEW source-status and rights qualifications.