  This paper constructs a kinematic basis for spin networks with planar or
cylindrical symmetry, by exploiting the fact that the basis elements are
representations of an O(3) subgroup of O(4). The action of the volume operator
on this basis gives a difference equation for the eigenvalues and eigenvectors
of the volume operator. For basis elements of low spin, the difference equation
can be solved readily on a computer. For higher spins, I solve for the
eigenvalues using a WKBJ method. This paper considers only the case where the
gravitational wave can have both polarizations. The single polarization case is
considered in a spearate paper.
