The work by N. Hitchin in 1987 opened a good possibility of describing the cotangent bundle of the moduli space of stable vector bundles over a compact Riemann surface in an explicit way. He proved that the space can be foliated by a family of certain spaces, i.e., the Jacobi varieties of spectral curves. The main purpose of this dissertation is to make the realization of the Hitchin system in a concrete way in the method initiated by I. M. Krichever and to give the necessary and sufficient condition for the linearity of flows in a Lax representation in terms of cohomological classes using the similar technique and analysis from the work by P. A. Griffiths.