Dynamical systems embedded into Lie algebras (Q2774766)

This article deals with the situation, where for a dynamical system \(\dot x= X(x)\) in \(\mathbb{R}^3\) the vector field \(X\) of the r.h.s. is embedded in the two-dimensional Lie algebra \(A_{22}\), NEWLINE\[NEWLINE[X, S_1]= a_0X+ a_1S_1,\quad a_0,a_1\in \mathbb{R},NEWLINE\]NEWLINE or in the three-dimensional Lie algebra \(A_{33}\), NEWLINE\[NEWLINE\begin{aligned} [X, S_1] &= a_0X+ a_2S_1+ a_2S_2,\\ [X, S_2] &= b_0X+ b_1S_1+ b_2 S_2,\\ [S_1, S_2] &= c_0 X+ c_1S_1+ c_2S_2.\end{aligned}NEWLINE\]NEWLINE It is shown and illustrated by examples that, under special additional conditions, one gets information on the first integrals, invariant foliations or integrability via quadratures. The question to decide whether or not such an embedding is possible can not answered completely.