Stäckel problem for non-diagonal Killing tensors: Yano-Patterson lifts, algebra of strong symmetries and quadratic in momenta integrals

TL;DR

The paper extends the Stäckel framework to non-diagonal Killing tensors by leveraging gl-regular Nijenhuis operators and Yano-Patterson lifts, constructing integrable systems with $n$ quadratic-in-momenta integrals that Poisson-commute. Central to the method is the symmetry algebra $\operatorname{Sym} L$ and the complete lift $\widehat L$, which yield a rich family of strong symmetries and a canonical symmetry $P$, enabling a broad class of Poisson-commuting integrals parameterized by $n(n+1)$ functions of two variables. In the diagonal case, the construction reduces to classical Stäckel separability; in dimension two it recovers all such systems, while higher dimensions furnish many new examples, including non-diagonal and Jordan-type Killing tensors with changing Segre characteristics. The authors also connect these finite-dimensional integrable systems to infinite-dimensional hydrodynamic-type PDEs $u_t = K_\alpha(u)u_x$ and demonstrate that, via generalised reciprocal transformations, the systems can be reduced to constant-coefficient form, highlighting both the analytical and geometric depth of the approach.

Abstract

We construct integrable Hamiltonian systems such that functionally independent Poisson commuting integrals are quadratic in the momenta. Unlike the classical Stäckel setting, we allow the associated self-adjoint $(1,1)$-tensors $K_α$ to be non-diagonalisable and have Jordan blocks and points where the Segre characteristic changes. Our construction is covariant and is based on Nijenhuis geometry: starting from a gl-regular Nijenhuis operator $L$ and its symmetry algebra, we obtain a large class of such integrable systems in a coordinate-free and signature-independent way; it is explicit once we have chosen a gl-regular Nijnhuis operator. In the diagonalisable case, our construction reproduces the Stäckel construction, and in dimension $n=2$ it recovers all known systems of this type; for $n\ge 3$ most of our systems are new. Finally, we establish applications to infinite-dimensional integrable systems of hydrodynamic type: namely, we show that for Killing $(1,1)$-tensors $ K_α$ corresponding to our example the evolutionarly PDE system of hydrodynamic type $u_t = K_α(u)u_x$ is integrable. We describe its symmetries, and use generalised reciprocal transformations to reduce it to a system with constant coefficient matrices.

Theorems & Definitions (43)

• Example 2.1: Strong symmetries of a complex structure
• Remark 2.1
• Theorem 2.1
• Corollary 2.1
• Remark 2.2
• proof : Proof of Theorem \ref{['t1']}
• Lemma 2.1
• proof
• Lemma 2.2
• proof