  The notion of Poisson quasi-Nijenhuis manifold generalizes that of
Poisson-Nijenhuis manifold. The relevance of the latter in the theory of
completely integrable systems is well established since the birth of the
bi-Hamiltonian approach to integrability. In this note, we discuss the
relevance of the notion of Poisson quasi-Nijenhuis manifold in the context of
finite-dimensional integrable systems. Generically (as we show by an example
with $3$ degrees of freedom) the Poisson quasi-Nijenhuis structure is largely
too general to ensure Liouville integrability of a system. However, we prove
that the closed (or periodic) $n$-particle Toda lattice can be framed in such a
geometrical structure, and its well-known integrals of the motion can be
obtained as spectral invariants of a "quasi-Nijenhuis recursion operator", that
is, a tensor field $N$ of type $(1,1)$ defined on the phase space of the
lattice. This example and some of its generalizations are used to understand
whether one can define in a reasonable sense a notion of {\em involutive\}
Poisson quasi-Nijenhuis manifold. A geometrical link between the open (or non
periodic) and the closed Toda systems is also framed in the context of a
general scheme connecting Poisson quasi-Nijenhuis and Poisson-Nijenhuis
manifolds.
