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Transparent pairs

Let $M$ be a closed orientable Riemannian surface. Consider an SO(3)-connection $A$ and a Higgs field $Φ:M\to so(3)$. The pair $(A,Φ)$ naturally induces a cocycle over the geodesic flow of $M$. We classify (up to gauge transformations) cohomologically trivial pairs $(A,Φ)$ with finite Fourier series in terms of a suitable Bäcklund transformation. In particular, if $M$ is negatively curved we obtain a full classification of SO(3)-transparent pairs.