  In \cite{kri02}, I. M. Krichever invented the space of matrices parametrizing
the cotangent bundle of moduli space of stable vector bundles over a compact
Riemann surface, which is named as the Hitchin system after the investigation
\cite{hit87}. We study a necessary and sufficient condition for the linearity
of flows on the space of Krichever-Lax matrices in a Lax representation in
terms of cohomological classes using the similar technique and analysis from
the work \cite{grif85} by P. A. Griffiths.
