Let $V: \mathbb{R}^n \to \mathbb{R}^n$ be a vector field that satisfies
\begin{equation} V(Rx) = RV(x) \qquad \forall R \in G \subset GL_n \end{equation}
where $G= O(n) $ or $G= GL_n$ (i.e. it is invariant under the group action on $V$). Can one infer a particular form of $V$? For example can one infer that
\begin{equation} V(x) = x f(||x||) \end{equation} holds ? At least for $n=2$, this is not necessarily true, but for $n>2$ I don't know, and have not found the right labels or tags to look after this question.
The label/ term that I was looking for was "Representations of isotropic tensor fields" and is answered in Representations of certain isotropic tensor functions. Roughly the argument is as follows:
Let $x \in \mathbb{R}^d$. Consider tha stabilizer sub group $G_x = \{ R \in G| Rx = x\} \subset G$. Then $\forall R \in G_x$
\begin{equation} RV(x) = V(Rx) = V(x) \end{equation}
by isotropy of $V$. Then if there exists $R \in G_x \setminus \{ id \} $ one could argue that the eigenspace $E_\lambda$ of that $R$ for $\lambda = 1$ is one dimensional and hence $V(x) = f(|| x||) x $. Alternatively by the above it follows that $G_x = G_{V(x)} =S G_x S^{-1}$ where $S x = V(x)$ and conclude that if $G_x \neq \{ id \} $ it necessarily follows that $S = f(||x||) x $.