  The Grassmannian model represents harmonic maps from Riemann surfaces by
families of shift-invariant subspaces of a Hilbert space. We impose a natural
symmetry condition on the shift-invariant subspaces that corresponds to
considering an important class of harmonic maps into symmetric and
$k$-symmetric spaces. In particular, we obtain new general forms for such
symmetric shift-invariant subspaces and for the corresponding extended
solutions.
