Bundle bispectrality for matrix differential equations

We consider the fundamental solutions of a wide class of first order system s with polynomial dependence on the spectral parameter and rational matrix potentials. Such matrix potentials are rational solutions of a large class of integrable nonlinear equations, which play an important role in differen t mathematical physics problems. The concept of bispectrality, which was or iginally introduced by Grunbaum, is extended in a natural way for the syste ms under consideration and their bispectrality is derived via the represent ation of the fundamental solutions. This bispectrality is preserved under t he flows of the corresponding integrable nonlinear equations. For the case of Dirac type (canonical) systems the complete characterization of the bisp ectral potentials under consideration is obtained in terms of the system's spectral function.