Non-invertible quasihomogeneous singularities and their Landau-Ginzburg orbifolds

According to the classification of quasihomogeneus singularities, any polynomial $f$ defining such singularity has a decomposition $f = f_κ+ f_{add}$. The polynomial $f_κ$ is of the certain form while $f_{add}$ is only restricted by the condition that the singularity of $f$ should be isolated. The polynomial $f_{add}$ is zero if and only if $f$ is invertible, and in the non-invertible case $f_{add}$ is arbitrary complicated. In this paper we investigate all possible polynomials $f_{add}$ for a given non-invertible $f$. For a given $f_κ$ we introduce the specific small collection of monomials that build up $f_{add}$ such that the polynomial $f = f_κ+ f_{add}$ defines an isolated quasihomogeneus singularity. If $(f,\mathbb{Z}/2\mathbb{Z})$ is Landau-Ginzburg orbifold with such non-invertible polynomial $f$, we provide the quasihomogeneus polynomial $\bar{f}$ such that the orbifold equivalence $(f,\mathbb{Z}/2\mathbb{Z}) \sim (\bar{f}, \{id\})$ holds. We also give the explicit isomorphism between the corresponding Frobenius algebras.