neutral element; and $\xi(t),\eta(t)$ are given curves on the tangent Lie algebra ${\mathfrak{g}}$ of $G$. Before proceeding note that, if $g(t)$ solves , then $g^{-1}(t)$ solves  for $\eta(t)=-\xi(t)$.

Such equations are pervasive in mathematics, physics and engineering. To begin with, $G$ can have a faithful finite-dimensional representation. For instance, consider (affine) linear differential equations on ${\mathbb{R}}^n$, $$\dot x = A(t)x + b(t) {\quad\mbox{with}\quad} x(t_0) = x_0.
\label{eq:manes-de-afin}$$ They are exactly solved by $$x(t) = {\mathcal{G}}(t,t_0)x_0 + \int_{t_0}^t{\mathcal{G}}(t,t')b(t')\,dt';$$ where the *Green’s function* ${\mathcal{G}}(t,t_0)$ is the matrix satisfying $$\frac{d{\mathcal{G}}(t,t_0)}{dt}\,{\mathcal{G}}^{-1}(t,t_0) = A(t), \quad {\mathcal{G}}(t_0,t_0)=1_n$$ of the kind . More generally, dynamical systems admitting a superposition principle can be reduced to the form , with $G$ a finite dimensional Lie group. This assertion is part of the classical *Lie–Scheffers* theory [@NihilNovumSubSole], reviewed in Section 3 as part of and motivation for the whole enterprise. Even more generally, any non-autonomous dynamical system, given in local coordinates by $$\frac{dx^i}{dt} = Y^i(t;x(t)), \qquad x = (x^1,\ldots,x^n)
\label{eq:alpha-and-omega}$$ corresponds to a ‘time-dependent vector field’ $Y$ with $Y(t)$ belonging to ${\mathfrak{X}}(M)$, the Lie algebra of all vector fields on a manifold $M$. Then the solution of  is given by the solution of an equation like ; this remark will 