Poisson quasi-Nijenhuis manifolds, closed Toda lattices, and generalized recursion relations

We present two involutivity theorems in the context of Poisson quasi-Nijenhuis manifolds. The second one stems from recursion relations that generalize the so-called Lenard–Magri relations on a bi-Hamiltonian manifold. We apply these results to the closed (or periodic) Toda lattices of type An(1), Cn(1), A2n(2), and, for the ones of type An(1), we show how this geometrical setting relates to their bi-Hamiltonian representation and to their recursion relations.