*(English)*Zbl 1091.34018

A system of differential equations

is said to admit a superposition principle if its general solution $y$ can be expressed as $y={\Phi}({y}_{\left(1\right)},\cdots ,{y}_{\left(m\right)};{k}_{1},\cdots ,{k}_{n})$ where $\{{y}_{\left(j\right)};\phantom{\rule{4pt}{0ex}}j=1,\cdots ,m\}$ is a set of independent particular solutions and ${k}_{1},\cdots ,{k}_{n}$ are $n$ arbitrary constants. Due to Sophus Lie, it is known that this is the case if and only if the system can be written in the form

where ${Z}_{1},\cdots ,{Z}_{r}$ being $r$ functions of only $x$, and ${\xi}^{\alpha i}$, $\alpha =1,\cdots ,r$, are functions of the variables $y=({y}^{1},\cdots ,{y}^{n})$, such that the $r$ vector fields in ${\mathbb{R}}^{n}$, given by

close on a finite-dimensional Lie algebra and $r\le mn$. The authors show that the study of such systems can be reduced to that of an equation on a Lie group, and that all such systems can be seen as the systems determining the horizontal curves on an appropriate connection. Some applications to the general Riccati equation and to quantum mechanics are given, too.