$\pi:E\to
M$. Vector fields along $f$ can also be regarded as $f$-derivations, in an obvious sense. The right hand side of the non-autonomous dynamical system  is just the vector field along the map $\pi_2:{\mathbb{R}}_t{\times}M\to M$ expressed in local coordinates by $$Y = Y^i(t;x(t))\,{\partial}_i\circ\pi_2.$$ Also, clearly any curve $\gamma:{\mathbb{R}}_t\to M$ defines a vector field $\dot\gamma$ along $\gamma$. We envisage here the concept of integral curves of vector fields $Y$ along maps $f$. These are curves $\gamma:{\mathbb{R}}_t\to N$ such that the image under $Tf\circ T\gamma$ of the vector field $d/dt$ coincides with the vector field along $f\circ\gamma$ given by $Y\circ\gamma$ —depending on $f$, there might be vector fields along it without integral curves. Under this definition $t\mapsto(t,{\gamma}(t))$ is always the integral curve of $(t,{\gamma}(t))\mapsto\dot\gamma(t)$ along $\pi_2:{\mathbb{R}}_t{\times}M\to M$.

The Lie–Scheffers theorem
=========================

The system  of differential equations admits a *superposition principle* —or possesses a set of fundamental solutions— if a superposition function $\Psi:{\mathbb{R}}^{n(m+1)}\to{\mathbb{R}}^n$ exists, written $$x = \Psi(x_{(1)},\ldots,x_{(m)};k_1,\ldots,k_n),$$ such that the general solution of  can be expressed (at least for small $t$) as the functional $$x(t) = \Psi(x_{(1)}(t),\dots,x_{(m)}(t);k_1,\dots,k_n),
\label{eq:manes-de-LS}$$ where ${\{\,x_{(a)}:a=1,\dots,m\,\}}$ is a set of particular solutions and $k_1,\dots,k_n$ denote $n$ arbitrary parameters. The latter must be *essential* in the sense that they 