Consider Maxwell's equation (without source):
$$ \partial_\mu F^{\mu \nu} = 0 \implies \partial_\mu \partial^\mu A^\nu = \partial_\mu \partial^\nu A^\mu.$$
Can we find a pair of classical field configurations $A^\mu(x),A'^\mu(x)$ such that they both satisfy the equation above (assuming similar boundary conditions) but are not related to each other by a gauge transformation of the type:
$$A'^\mu(x) = A^\mu(x)+\partial^\mu \varphi(x) \quad ?$$
If it's impossible, how could we argue/show this?
Answer: Thanks to my2cts' answer, I've found 2 solutions $A$ and $A'$ not related by a gauge transformation : $A^\mu=(0,e^{-i(t-y)},0,0)$ and $A'^\mu=(0,0,e^{-i(t-x)},0)$. It makes sense since they both give rise to different EM fields, which are invariant under gauge transformations.
