Section 12 explores by means of pre-Lie algebras with Rota–Baxter maps the solution of that nonlinear recursion in the general case. Section 13 is the conclusion, whereupon perspectives for research are discussed.

As said, Appendix A reviews the basics of Lie group and Lie algebra actions on manifolds. As also hinted at, Appendices B and C run a parallel, complementary strand to the main body of the paper. They contain more advanced material on the topic of dynamical systems with symmetry; their treatment here naturally calls for the Darboux derivative of Lie algebroid theory. The main point is to show how one is led to the arena of Lie algebra and *geometrical integration*, for the solution of differential equations we are concerned with. This provides a rationale for our choice of the Magnus series, and its generalizations, as the primary approximation method in the body of the paper —see the discussion at the beginning of Section 10.

Appendix D gives the Hopf algebraic vision of a theorem of Lie and Engel.

Several notational conventions are fixed next. Let $M$ be a (second countable, smooth, without boundary) manifold of finite dimension $n$. The space ${\mathcal{F}}(M)$ of (real or) complex smooth functions 