Abstract
The theory of Dimakis and Müller-Hoissen (Dimakis A and Müller-Hoissen F 2000 J. Phys. A: Math. Gen. 33 957-74) concerning bi-differential calculi and completely integrable systems is related to bi-Hamiltonian systems of the Poisson-Nijenhuis type. In the special case where the ambient manifold is a cotangent bundle one is able to recover and elucidate the theory of Ibort et al (Ibort A, Magri F and Marmo G 2000 J. Geom. Phys. 33 210-23), which is in turn a reworking in the bi-Hamitonian context of Benenti's theory of Hamilton-Jacobi separable systems. In particular, it is shown that Benenti's conformal Killing tensor, which is central to his theory, has an even more special form than has hitherto been realized and that when it is converted into a field of endomorphisms by raising an index with the ambient metric, it necessarily has vanishing Nijenhuis torsion.
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