  A previous paper constructed a kinematic basis for spin networks with planar
or cylindrical symmetry and arbitrary polarization. This paper imposes a
constraint which limits the gravitational wave to a single polarization. The
spectrum of the constraint contains a physically reasonable number of zero
eigenvalues, and the zero eigenvectors can be constructed explicitly.
Commutation of the constraint with the Hamiltonian is expected to lead to a
further constraint. This new constraint is not investigated in this paper, but
I argue it will be non-local, relating states at two or more neighboring
vertices.
