Preprint A40/2001
Bundle Bispectrality for Matrix Differential Equations

A.L. Sakhnovich | Zubelli, Jorge P.

**Keywords: **
bispectrality | nonlinear evolution equations | integrable

We consider the fundamental solutions of a wide class of first
order systems 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 different mathematical physics problems.
The concept of bispectrality, which was originally
introduced by Grúnbaum, is extended in a natural way for
the systems under consideration and their bispectrality is derived via
the representation of the fundamental solutions. This bispectrality is
preserved under the flows of the corresponding integrable nonlinear equations.
For the case of Dirac type (canonical) systems the complete characterization
of the bispectral potentials under consideration is obtained in terms of
the system's spectral function.