The flexibility of $m$-suspensions constructed via a local slice

We refer to the variety $\mathrm{Susp}(X,f,k_1,\dots ,k_m)=\mathbb{V}(y_1^{k_1}\dots y_m^{k_m}-f(x))\subset X\times {\mathbb{A}}^m$ as an $m$-suspension over affine variety $X$, constructed via a local slice $f(x)\in \mathbb{K}[X]$, if there exists a locally nilpotent derivation $\delta$ on $X$ such that $\delta (f)\mathrm{\ne }0,{\delta }^2(f)=0$. In this paper, we determine the sufficient conditions under which such a variety is generically flexible and those under which it is flexible. Furthermore, for a flexible $X$ we propose a construction of a local slice $f$ that guarantees the flexibility of the suspension $\mathrm{Susp}(X,f,1,k_2,\dots ,k_m)$.