On three-dimensional Poisson quasi-Nijenhuis manifolds and Haantjes structures

In this note we first characterize Poisson quasi-Nijenhuis structures on three-dimensional oriented manifolds whose underlying Poisson tensor never vanishes.

We then apply this result to show that each of these structures is (locally) a deformation of a PN structure and is involutive.

Finally, we prove that every such three-dimensional Poisson quasi-Nijenhuis manifold is a Haantjes manifold and that it carries a generalized Lenard-Magri chain.