A new finite-dimensional Hamiltonian systems with a mixed Poisson structure for the KdV equation

A Lax pair for the KdV equation is derived by a transformation of the eigenfunction. By a polynomial expansion of the eigenfunction for the resulting Lax pair, finite-dimensional integrable systems can be obtained from the Lax pair. These integrable systems are proved to be the Hamiltonian and are shown to have a new Poisson structure such that the entries of its structure matrix are a mixture of linear and quadratic functions of coordinates. The odd and even functions of the spectral parameter are introduced to build a generating function for conserved integrals. Based on the generating function, the integrability of these Hamiltonian systems is shown.