  Using $\mathbb{Z}_3$ symmetry, we present a topological condition for the
existence of the $\mathbb{Z}_2$ harmonic 1-forms over Riemannian manifold. As a
corollary, if $L$ is an oriented link on $S^3$ with determinant zero, then
there exists a nondegenerate $\mathbb{Z}_2$ harmonic 1-form over the 3-cyclic
branched covering of $L$. Furthermore, we found infinite number of rational
homology 3-spheres that admit a nondegenerate $\mathbb{Z}_2$ harmonic 1-form.
