Let $C$ be a complex curve. Recall that a Higgs bundle on $C$ is a vector bundle $E$ on $C$ equipped with a morphism $E \to E \otimes K\_C$. The space of (stable) Higgs bundles is much studied, and is in particular known to be smooth. Moreover there is a "nonabelian Hodge theorem" giving a diffeomorphism between the moduli of Higgs bundles and a certain character variety of $\pi\_1(C)$. 



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> What is known about the moduli space of Higgs bundles with a section, i.e., the space parameterizing triples $(E, s \in \mathrm{H}^{0}(E), \phi: E\to E \otimes K\_C)$ ? Is it smooth (after imposing some appropriate stability condition)? Is there an analogue of the "nonabelian Hodge theorem"?
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What is known about Higgs bundles with sections?
There is a fundamental difference between the case of Higgs bundles (where the section lies in a twisted adjoint representation) and the case of a section of the bundle itself (where the section is in the vector representation). In the former, the notion of stability is rigid, whereas in the latter the definition of stability depends on a parameter. This was discovered by Bradlow (J. Differential Geom. 33 (1991), no. 1, 169--213) and Bradlow-Daskalopoulos (Internat. J. Math. 2 (1991), no. 5, 477--513) and exploited by Thaddeus (Invent. Math. 117, no. 2 (1994), 317--353). These papers will point you in the direction of a definition of stability/semistability for the case you're interested in. I guess it will be true that stable points are smooth, though for certain values of the parameter the compactifications will contain strictly semistable (non-smooth) points.


To answer your question, there is apparently no relationship between Bradlow pairs and representations of the fundamental group. Rather, these spaces are more closely related to higher rank generalizations of symmetric products of the Riemann surface (see also J. Amer. Math. Soc. 9 (1996), 529-571). 


