  We construct a class of matrix-valued Schr\"odinger operators with prescribed
finite-band spectra of maximum spectral multiplicity. The corresponding matrix
potentials are shown to be stationary solutions of the KdV hierarchy. The
methods employed in this paper rely on matrix-valued Herglotz functions,
Weyl--Titchmarsh theory, pencils of matrices, and basic inverse spectral theory
for matrix-valued Schr\"odinger operators.
