We use local symplectic Lie groupoids to construct Poisson
integrators for generic Poisson structures. More precisely, recursively ob-
tained solutions of a Hamilton-Jacobi-like equation are interpreted as La-
grangian bisections in a neighborhood of the unit manifold, that, in turn,
give Poisson integrators. We also insist on the role of the Magnus for-
mula, in the context of Poisson geometry, for the backward analysis of
The talk is based on the preprint https://arxiv.org/abs/2205.04838.