when can one find a $G$-connection that makes a particular section constant

Question

Let $M$ be a smooth manifold and let $E=\mathcal{O}_M^n$ be a trivial vector bundle. Assume that a linear group $G \subset \mathrm{GL}_n$ acts on $E$ naturally. Let $s$ be a section $s \in H^0(M, E)$. What is the necessary and sufficient condition for existence of a section $a \in H^0(M, LG \otimes \Omega_M^1)$ such that $a \cdot s = -ds$, where $ds$ is understood as section of $E \otimes \Omega^1_M$, and $LG \subset \mathrm{End}(\mathbb{R}^n)$ is the Lie algebra of $G$?

Answer 1

Formulating the question precisely helped me to understand what is happening: I will post an answer myself.

Substititing vector fields in the 1-forms that are part of the question, one finds that it is really equivalent to the following one. Let $G$ be a liner group acting naturally on $\mathbb{R}^n$, and let $x(t)$ be a curve in $\mathbb{R}^n$. What is the necessary and sufficient condition that for any $t$ there exists an endomorphism $a$ from the Lie algebra $LG$ (acting on the tangent space to $\mathbb{R}^n$ which we identify with $\mathbb{R}^n$ itself) such that $a(x(t)) = -\dot{x}(t)$, where $\dot{x}(t)$ is the tangent vector? This is clearly true if the linear subspace $LG \cdot x(t) \subset \mathbb{R}^n$ is constant for all $t$, then clearly $-\dot{x}(t)$ lies in the same space, and, tautologically, the required endomorphism from $LG$ can be found.