Lemma 2.8 of this paper https://link.springer.com/article/10.1007/BF01232026?noAccess=true says
For any $x \in X$ trajectory of $x$ by the gradient flow $\nabla f $ lies in the $G$-orbit $\mathcal{O}=Gx.$
here $X$ is a manifold on which acts a group $G$, and $f$ is a smooth function on $X$.
I didn't understand the statement of this lemma; what it means by the trajectory of a point by the flow of $\nabla f$ ?
I believe that the trajectory of $x$ through $\nabla f$ is obtained through the flow of $\nabla f$, seen as a vector field on $X$, starting at point $x$, so the unique $t \mapsto \tau(t)$ such that $\tau(0) = x$ and $\tau'(t) = \nabla f(\tau(t))$ for each $t$ (at least in an interval around $0$). I think you need additional assumptions to have the result you quote (some sort of link between $f$ and $G$ : else, choose $G = \{ e \}$ the trivial group and $f$ any function such that $\nabla f(x) \neq 0$). This should appear in the paper.
(Unfortunately, I cannot access to the paper you linked to without having to pay a subscription, so I hope my guess is correct.)