  The goal of this paper is to give an explicit description of the integrable
structure of the Hitchin moduli spaces. This is done by introducing explicit
parameterisations for the different strata of the Hitchin moduli spaces, and by
adapting the Separation of Variables method from the theory of integrable
models to the Hitchin moduli spaces. The resulting description exhibits a clear
analogy with Drinfeld's first construction of the geometric Langlands
correspondence. It can be seen as a classical limit of a version of Drinfeld's
construction which is adapted to the complex number field.
