Let $X$ be a (non-singular) complex surface and $(V,\nabla)$ be a vector bundle $V$ equipped with a flat connection $\nabla$ on $X$. Fix a point $x \in X$ and $v_0 \in V_x$ an element in the fiber over the point $x$ of the vector bundle $V$. Does there exist a global section $s$ of the vector bundle $V$ such that at the point $x$, it takes the value $v_0$?

Let $X$ be a (non-singular) complex surface and $(V,\nabla)$ be a vector bundle $V$ equipped with a flat connection $\nabla$ on $X$. Fix a point $x \in X$ and $v_0 \in V_x$ an element in the fiber over the point $x$ of the vector bundle $V$. Does there exist a global section $s$ of the vector bundle $V$ such that at the point $x$, it takes the value $v_0$? I think this is true if $X$ is simply connected.