The coupled curvature equations

Question

Let $E$ be a vector bundle with two connections $\nabla =d+A$ and $\nabla'=d+A'$. Then, I define coupled curvature equations:

$$dA+ A\wedge A= dA'+ A' \wedge A'$$

$$dA+dA'+A \wedge A' + A' \wedge A=0$$

The gauge group acts on these equations. Over the trivial bundle over a Riemann surface, can we act on the metric connections by the gauge group (we take the parts of type $(0,1)$) to obtain solutions of these coupled equations from any two metric  connections?