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Minimal and CMC surfaces in $S^3$ can be treated via their associated familyof flat $\SL(2,\C)$-connections. In this the paper we parametrize the modulispace of flat $\SL(2,\C)$-connections on the Lawson minimal surface of genus 2which are equivariant with respect to certain symmetries of Lawson's geometricconstruction. The parametrization uses Hitchin's abelianization procedure towrite such connections explicitly in terms of flat line bundles on a complex1-dimensional torus. This description is used to develop a spectral curvetheory for the Lawson surface. This theory applies as well to other CMC andminimal surfaces with the same holomorphic symmetries as the Lawson surface butdifferent Riemann surface structure. Additionally, we study the space ofisospectral deformations of compact minimal surface of genus $g\geq2$ and provethat it is generated by simple factor dressing.

A spectral curve approach to Lawson symmetric CMC surfaces of genus 2