Dressing for generalised linear Hamiltonian systems depending rationally on the spectral parameter and some applications

Abstract

We construct so called Darboux matrices and fundamental solutions in the important case of the generalised Hamiltonian (or canonical) systems depending rationally on the spectral parameter. A wide class of explicit solutions is obtained in this way. Interesting results for dynamical systems depending on several variables and their explicit solutions follow. For these purposes we use our version of B\"acklund-Darboux transformation and square roots of the corresponding generalised matrix eigenvalues. Some new auxiliary results on the roots of matrices are included as well. An appendix is added to make the paper self-sufficient.