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.