  This paper addresses a gap in the classifcation of Codazzi tensors with
exactly two eigenfunctions on a Riemannian manifold of dimension three or
higher. Derdzinski proved that if the trace of such a tensor is constant and
the dimension of one of the the eigenspaces is $n-1$, then the metric is a
warped product where the base is an open interval- a conclusion we will show to
be true under a milder trace condition. Furthermore, we construct examples of
Codazzi tensors having two eigenvalue functions, one of which has eigenspace
dimension $n-1$, where the metric is not a warped product with interval base,
refuting a remark in \cite{Besse} that the warped product conclusion holds
without any restriction on the trace.
