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The goal of this paper is to give an explicit description of the integrablestructure of the Hitchin moduli spaces. This is done by introducing explicitparameterisations for the different strata of the Hitchin moduli spaces, and byadapting the Separation of Variables method from the theory of integrablemodels to the Hitchin moduli spaces. The resulting description exhibits a clearanalogy with Drinfeld's first construction of the geometric Langlandscorrespondence. It can be seen as a classical limit of a version of Drinfeld'sconstruction which is adapted to the complex number field.

Classical limit of the geometric Langlands correspondence for $SL_2(\mathbb{C})$