I am reading a little about magnetic monopoles, with gauge group $SU(2)$. In the book "The Geometry and Dynamics of Magnetic Monopoles", Atiyah and Hitchin consider a pair $(A,\phi)$ where $A$ is a connection on a principal $SU(2)$ bundle on a domain in $\mathbb{R}^3$, and $\phi$ is a section of the adjoint bundle.
At some point, the authors write "use the gauge $A_1 = 0$". My question is about that. They seem to be saying that one can always find a gauge transformation, call it $g$, which can be thought of as a smooth function from the domain in $\mathbb{R}^3$ into $SU(2)$, such that
$-dgg^{-1} + gAg^{-1}$
has no $dx$ component, which amounts to finding a gauge transformation $g$ such that
$g^{-1} \frac{\partial g}{\partial x} = A_1$.
My first attempt was to write
$g = e^{\int A_1 dx} g_0$
but when attempting to check if this was indeed a solution, I got stuck since $A_1$ and $\int A_1 dx$ do not commute in general, right?
However I vaguely remember reading something like this in a physics book, where I think Dyson solved it using some kind of ordering on operators. Or am I confusing two different things? Could someone please indicate how to solve the differential equation above please?
Edit 1: this seems to be related to the Peano-Baker series or, on the Physics side, the Dyson series, with time-ordered (in my case $x_1$-ordered) products. Ok, I think I know where to look for an answer now. However, if someone wants to write an answer, then that would be good too. In particular, what assumptions on $A_1$ guarantee that there exists at least one smooth solution?