### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 12 (2016), 024, 4 pages      arXiv:1502.07516      https://doi.org/10.3842/SIGMA.2016.024
Contribution to the Special Issue on Analytical Mechanics and Differential Geometry in honour of Sergio Benenti

### Nijenhuis Integrability for Killing Tensors

Mathematisches Institut, Fakultät für Mathematik und Informatik, Friedrich-Schiller-Universität Jena, 07737 Jena, Germany

Received October 30, 2015, in final form February 26, 2016; Published online March 07, 2016

Abstract
The fundamental tool in the classification of orthogonal coordinate systems in which the Hamilton-Jacobi and other prominent equations can be solved by a separation of variables are second order Killing tensors which satisfy the Nijenhuis integrability conditions. The latter are a system of three non-linear partial differential equations. We give a simple and completely algebraic proof that for a Killing tensor the third and most complicated of these equations is redundant. This considerably simplifies the classification of orthogonal separation coordinates on arbitrary (pseudo-)Riemannian manifolds.

Key words: integrable systems; separation of variables; Killing tensors; Nijenhuis tensor; Haantjes tensor.

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