can be solved from the solution functional: $$k = \Xi\bigl(x_{(1)}(t),\dots,x_{(m)}(t);x(t)\bigr), {\quad\mbox{with}\quad} k
:= (k_1,\dots,k_n).
\label{eq:desmanes-de-LS}$$

The Lie–Scheffers theorem [@NihilNovumSubSole] asserts:

For  to admit a superposition principle it is necessary and sufficient that the time-dependent vector field $Y$ be of the form $$Y(t;x) = Z_1(t)X_1(x) + \cdots + Z_r(t)X_r(x),
\label{eq:cascabel-al-gato}$$ where, as indicated in the notation, the $r$ scalar functions $Z_a$ depend only of $t$ and the $r$ vector fields $X_a$ depend only on the variables $x$; and these fields close to a real Lie algebra ${\mathfrak{g}}$. That is, the $X_a$ are linearly independent and there exist suitable structure constants $f_{ab}^c$ such that $$[X_a, X_b] = \sum_{c=1}^r f_{ab}^c\,X_c.$$ Moreover, the dimension $r$ of ${\mathfrak{g}}$ is not greater than $nm$. Systems fulfilling the conditions of the theorem are called here Lie–Scheffers systems associated to ${\mathfrak{g}}$.

Modern reviews of this subject include [@macrote; @TresMosqueteros; @Ciguenya; @PBPepinArturo; @TresMosqueterosbis]. We ponder a few pertinent examples of Lie–Scheffers systems next.

Linear systems  are Lie–Scheffers systems. If $n+1$ particular solutions $x_{(1)},\ldots,x_{(n+1)}$ of  are known, such that $x_{(2)}(t)-x_{(1)}(t),\ldots,x_{(n+1)}(t)- x_{(1)}(t)$ are independent, and $H(t)$ is the regular matrix with these vectors as columns, then the vector of parameters  is given by $$k = H^{-1}(t)\bigl(x(t) - x_{(1)}(t)\bigr).$$ This follows from 