On the classification of function germs of two variables that are equivariant simple with respect to an action of the cyclic group of order three

We consider the problem to classify function germs $(\mathbb{C}^2,0)\to(\mathbb{C},0)$ that are equivariant simple with respect to nontrivial actions of the group $\mathbb{Z}^3$ on $\mathbb{C}^2$ and on $\mathbb{C}$ up to equivariant automorphism germs $(\mathbb{C}^2,0)\to(\mathbb{C}^2,0)$. The complete classification of such germs is obtained in the case of nonscalar action of $\mathbb{Z}^3$ on $\mathbb{C}^2$ that is nontrivial in both coordinates. Namely, a germ is equivariant simple with respect to such a pair of actions if and only if it is equivalent to ine of the following germs:

\begin{eqnarray*} (x,y)&\mapsto& x^{3k+1}+y^2, \quad k\geqslant1;\\ (x,y)&\mapsto& x^2y+y^{3k-1}, \quad k\geqslant2;\\ (x,y)&\mapsto& x^4+xy^3;\\ (x,y)&\mapsto &x^4+y^5. \end{eqnarray*}