It is known that the KdV equation possesses 2 equivalent Hamiltonian formulations. One Poisson bracket is known as the Gardner-Zakharov-Faddeev bracket, and the other one is called the Magri bracket. Several Hamiltonian formulations are possible for most integrable equations with a Lax representation. All Hamiltonian structures arise from Krichever-Phong's universal formula.
In this talk we will discuss some new results in the finite-dimensional case, i.e., when the Lax function is a meromorphic matrix function either on the Riemann sphere, or on an elliptic curve. In particular, we show that a multiplicative representation linearizes the so-called quadratic Poisson bracket (the Magri bracket for the KdV). Interestingly, this result is parallel to the Kupershmidt-Wilson theorem in the infinite-dimensional case. The results may be found in the paper: http://arxiv.org/abs/0811.3784