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.

中文翻译：

---

Poisson 拟-Nijenhuis 流形和 Toda 系统

Poisson quasi-Nijenhuis 流形的概念推广了 Poisson-Nijenhuis 流形。后者在完全可积系统理论中的相关性自双汉密尔顿可积性方法诞生以来就已确立。在本笔记中，我们讨论了泊松拟 Nijenhuis 流形概念在有限维可积系统的上下文中的相关性。一般而言（正如我们通过具有 $3$ 自由度的示例所示），Poisson quasi-Nijenhuis 结构在很大程度上过于笼统，无法确保系统的 Liouville 可积性。然而，我们证明了封闭（或周期性）$n$-粒子 Toda 晶格可以在这样的几何结构中构建，并且其众所周知的运动积分可以作为“准 Nijenhuis 递归算子的谱不变量”获得“， 那是，在晶格的相空间上定义的类型为 $(1,1)$ 的张量场 $N$。这个例子及其一些概括用于理解人们是否可以在合理的意义上定义 {\em involutive\} Poisson quasi-Nijenhuis 流形的概念。开放（或非周期性）和封闭 Toda 系统之间的几何联系也在连接 Poisson quasi-Nijenhuis 和 Poisson-Nijenhuis 流形的一般方案的背景下构建。