Bi-differential calculi, bi-Hamiltonian systems and conformal Killing tensors

The theory of Dimakis and Muller-Hoissen (Dimakis A and Muller-Hoissen F 20 00 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 i s in turn a reworking in the bi-Hamitonian context of Benenti's theory of H amilton-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.