Nonlinearization of spectral problems for the perturbation KdV systems

The theory of nonlinearization of spectral problems is developed for investigating the perturbation KdV systems, and a kind of finite-dimensional integrable Hamiltonian systems is generated from spectral problems of the perturbation KdV systems. To establish the Liouville integrability of the obtained Hamiltonian systems, we identify them with the perturbation systems of the Garnier system, which can be directly proved to be integrable. A useful formula to compute the functional gradient of the spectral parameter with respect to the potential is also presented for arbitrary-order matrix spectral problems.