Simplest singular points of $1$-forms invariant with respect to actions of a third-order group

We classify singular points which cannot be excluded by deformations of $1$-forms invariant with respect to an action of a cyclic group of order $3$. It is proved that for $\mathbb{Z}_3$-invariant $1$-forms the equivariant index of a singular point as an element of the representation ring of the group coincides with the class of the representation on the space of germs of the highest order forms factorized by the subspace of forms divisible by the given $1$-form.