  According to the classification of quasihomogeneus singularities, any
polynomial $f$ defining such singularity has a decomposition $f = f_\kappa +
f_{add}$. The polynomial $f_\kappa$ 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 $f_\kappa$ we introduce the specific small
collection of monomials that build up $f_{add}$ such that the polynomial $f =
f_\kappa + f_{add}$ defines an isolated quasihomogeneus singularity. If
$(f,\mathbf{Z}/2\mathbf{Z})$ is Landau-Ginzburg orbifold with such
non-invertible polynomial $f$, we provide the quasihomogeneus polynomial
$\bar{f}$ such that the orbifold equivalence $(f,\mathbf{Z}/2\mathbf{Z}) \sim
(\bar{f}, \{id\})$ holds. We also give the explicit isomorphism between the
corresponding Frobenius algebras.
