Let $A$ be an abelian variety over a field $k$ of characteristic $0$. How do I prove, without using transcendental methods, that if $\nabla$ is an integrable connection on a vector bundle $L$ on $A$ then $(L, \nabla)$ comes from a $G$-bundle with connection for some abelian algebraic subgroup $G \subset GL(n)$? And the corollary there exists a $\nabla$-stable flag$$L = L_n \supset L_{n-1} \supset \dots \supset L_1 \supset 0$$with $\text{rank }L_i = i$? It is easy to prove for $k = \mathbb{C}$ by looking at the monodromy of $(L, \nabla)$, and the general case follows by the Lefschetz principle, but I want to find a different proof which does not use $\mathbb{C}$ at all.
Any help would be greatly appreciated!