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

TL;DR

This paper develops a covariant, Nijenhuis geometry-based method to construct integrable Hamiltonian systems with quadratic integrals, extending classical Stäckel theory to non-diagonalisable tensors and revealing new systems, including applications to hydrodynamic PDEs.

Contribution

It introduces a novel, coordinate-free approach to integrable systems using Nijenhuis geometry, generalizing Stäckel constructions to non-diagonalisable tensors and providing new integrable models.

Findings

Reproduces classical Stäckel systems in the diagonalisable case

Most constructed systems are new for dimensions n ≥ 3

Establishes integrability of certain hydrodynamic type PDEs

Abstract

We construct integrable Hamiltonian systems such that functionally independent Poisson commuting integrals are quadratic in the momenta. Unlike the classical St\"ackel setting, we allow the associated self-adjoint $(1,1)$-tensors $K_\alpha$ 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 St\"ackel 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_\alpha$ corresponding to our example the evolutionarly PDE system of hydrodynamic type $u_t = K_\alpha(u)u_x$ is integrable. We describe its symmetries, and use generalised reciprocal transformations to reduce it to a system with constant coefficient matrices.