Lawson's genus two minimal surface and meromorphic connections

We investigate the Lawson genus $2$ surface by methods from integrable system theory. We prove that the associated family of flat connections comes from a family of flat connections on a $4-$punctured sphere. We describe the symmetries of the holonomy and show that it is already determined by the holonomy around one of the punctures. We show the existence of a meromorphic DPW potential for the Lawson surface which is globally defined on the surface. We determine this potential explicitly up to two unknown functions depending only on the spectral parameter.

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