Tyurin parameters of commuting pairs and infinite dimensional Grassmann manifold

Commuting pairs of ordinary differential operators are classified by a set of algebro-geometric data called algebraic spectral data''. These data consist of an algebraic curve (spectral curve’‘) $\mathrm{\Gamma }$ with a marked point ${\gamma }_{\mathrm{\infty }}$, a holomorphic vector bundle $E$ on $\mathrm{\Gamma }$ and some additional data related to the local structure of $\mathrm{\Gamma }$ and $E$ in a neighborhood of ${\gamma }_{\mathrm{\infty }}$. If the rank $r$ of $E$ is greater than 1, one can use the so called ``Tyurin parameters’’ in place of $E$ itself. The Tyurin parameters specify the pole structure of a basis of joint eigenfunctions of the commuting pair. These data can be translated to the language of an infinite dimensional Grassmann manifold. This leads to a dynamical system of the standard exponential flows on the Grassmann manifold, in which the role of Tyurin parameters and some other parameters is made clear.