the fact that the transformation $$x'(t) = H^{-1}\bigl(x(t) - x_{(1)}(t)\bigr)$$ reduces the system to $dx'/dt=0$.

A famous example for $n=1$ is provided by the Riccati equation: $$\dot x = a_0(t) + a_1(t)x + a_2(t)x^2.
\label{eq:manes-de-Riccati}$$ One can understand by Hopf algebraic methods why, up to diffeomorphisms, Riccati’s is the only nonlinear Lie–Scheffers differential equation on the real line; this was indicated in [@Bruna] and it is spelled in Appendix D. Also  is the simplest Lie–Scheffers system not solvable by quadratures; and other Lie–Scheffers equations on the line are reductions of it, in an appropriate sense —see Appendix B. The superposition principle for the Riccati equation is given by $$k = \frac{(x - x_{(2)})(x_{(1)} - x_{(3)})}{(x - x_{(1)})(x_{(2)} -
x_{(3)})}.$$

The one-dimensional example $$\dot x = b(t)\chi(x)
\label{eq:tugurito}$$ is instructive. We assume $\chi$ does not change sign in the interval of interest. Let $$\phi(x) := \int^x\frac{dx'}{\chi(x')}.$$ Then $$x(t) = \phi^{-1}\Bigl(k' + \int b(t)\,dt\Bigr)$$ is the general solution. We have therefore a superposition rule of the form $$x(t) = \phi^{-1}\bigl(\phi(x_{(1)}(t)) + k\bigr),$$ with $m=n=r=1$: only one particular solution is required. Notice that the local diffeomorphism $\phi$ projects the vector field corresponding to the right hand side in  to the vector field $b(t){\partial}_x$: 