Numerical flow-box theorems under structural assumptions

The numerical flow-box theorem says that locally, in the vicinity of nonequ ilibria, discretized solutions of an autonomous ordinary differential equat ion are exact solutions of a modified equation nearby: for stepsize h suffi ciently small the original discretization operator is the time-h map of the solution operator of the modified equation. It is shown that the very same result holds true in the following categories of differential equations an d discretizations: I/ preserving a finite number of first integrals; V/ preserving the volume form; S/ preserving the canonical symplectic form.