I recently came across a way to think of an ordinary differential equation on a smooth manifold $M$ is as a Lie group homomorphism $\phi : (\mathbb{R}, +) \rightarrow \mathrm{Diff}(M)$ where $\mathrm{Diff}(M)$ is the group of smooth diffeomorphisms of $M$.

If we substitute the additive group of reals with another Lie group $G$ in this point of view, we could regard the resulting map $\phi : G \rightarrow \mathrm{Diff}(M)$ as a "G-differential equation" or "G-flow" on $M$.

Do these kinds of flows naturally arise and are they well studied? Are there some nice examples of G-flows for classical groups?

I recently came across a way to think of an ordinary differential equation on a smooth manifold $M$ is as a Lie group homomorphism $\phi : (\mathbb{R}, +) \rightarrow \operatorname{Diff}(M)$ where $\operatorname{Diff}(M)$ is the group of smooth diffeomorphisms of $M$.

If we substitute the additive group of reals with another Lie group $G$ in this point of view, we could regard the resulting map $\phi : G \rightarrow \operatorname{Diff}(M)$ as a "$G$-differential equation" or "$G$-flow" on $M$.

Do these kinds of flows naturally arise and are they well studied? Are there some nice examples of $G$-flows for classical groups?