  Minimal and CMC surfaces in $S^3$ can be treated via their associated family
of flat $\SL(2,\C)$-connections. In this the paper we parametrize the moduli
space of flat $\SL(2,\C)$-connections on the Lawson minimal surface of genus 2
which are equivariant with respect to certain symmetries of Lawson's geometric
construction. The parametrization uses Hitchin's abelianization procedure to
write such connections explicitly in terms of flat line bundles on a complex
1-dimensional torus. This description is used to develop a spectral curve
theory for the Lawson surface. This theory applies as well to other CMC and
minimal surfaces with the same holomorphic symmetries as the Lawson surface but
different Riemann surface structure. Additionally, we study the space of
isospectral deformations of compact minimal surface of genus $g\geq2$ and prove
that it is generated by simple factor dressing.
