  We use local symplectic Lie groupoids to construct Poisson integrators for
generic Poisson structures. More precisely, recursively obtained solutions of a
Hamilton-Jacobi-like equation are interpreted as Lagrangian bisections in a
neighborhood of the unit manifold, that, in turn, give Poisson integrators. We
also insist on the role of the Magnus formula, in the context of Poisson
geometry, for the backward analysis of such integrators.
