Remarks on the eigenvalues of the Manakov system

We consider the Manakov system with L1 potentials q1 and q2 and study conditions under which the set of eigenvalues exhibits certain symmetry properties. In particular we obtain criteria which guarantee that the eigenvalues are located symmetrically with respect to the imaginary axis. We also obtain a best possible bound in terms of an integral involving the two potentials which ensures that no eigenvalues exist.