  We study the parametrization of the moduli space Bun_2(C)_L of rank 2 bundles
over a curve C with fixed determinant, provided by Hecke modifications at fixed
points of the trivial bundle. This parametrization is closely related to the
Tyurin parametrization of vector bundles over curves. We use it to parametrize
the Hitchin and KZB systems, as well as lifts of the Beilinson-Drinfeld
D-modules. We express a generating series for the lifts of the
Beilinson-Drinfeld operators in terms of a "quantum L-operator" \ell(z). We
explain the relation to earlier joint work with G. Felder, based on
parametrization by flags of bundles (math/9807145) and introduce filtrations on
conformal blocks, related with the Hecke modifications.
