We use the properties of $Mp\left(n\right)$ to construct functions ${\mu }_{\ell }:Mp\left(n\right)\to {\mathbf{Z}}_8$ associated with the elements $\ell$ of the lagrangian grassmannian $\Lambda$ (n) which generalize the Maslov index on Mp(n) defined by J. Leray in his “Lagrangian Analysis”. We deduce from these constructions the identity between $Mp\left(n\right)$ and a subset of $Sp\left(n\right)\times {\mathbf{Z}}_8$, equipped with appropriate algebraic and topological structures.

We discuss the role of Poisson-Nijenhuis (PN) geometry in the definition of multiplicative integrable models on symplectic groupoids. These are integrable models that are compatible with the groupoid structure in such a way that the set of contour levels of the hamiltonians in involution inherits a topological groupoid structure. We show that every maximal rank PN structure defines such a model. We consider the examples defined on compact hermitian symmetric spaces studied by F. Bonechi, J. Qiu...