[@BCT19] the authors considered the generating series $$F^{\mathrm{ext}}_{{\text{$r$-spin}}}(t^1,\ldots,t^r):=\sum_{n\ge 2}\frac{1}{n!}\sum_{0\le\alpha_1,\ldots,\alpha_n\le r-1}\left(\int_{{\overline{\mathcal M}}_{0,n+1}}W^r_{0,n+1}(-1,\alpha_1,\ldots,\alpha_n)\right)\prod_{i=1}^n t^{\alpha_i+1}$$ and proved that it satisfies the open WDVV equations, associated to the potential $F_{{\text{$r$-spin}}}$, together with property . Here one should identify $t^r=s$. It occurs that after a simple transformation the function $F^{\mathrm{ext}}_{{\text{$r$-spin}}}$ also gives the generating series of intersection numbers on the moduli space of $r$-spin disks, introduced in [@BCT18 Theorem 1.3].

Introduce a function $F^o_{A_N}(t^1,\ldots,t^N,s)$ by $$F^o_{A_N}(t^1,\ldots,t^N,s):=(-r)^{-2}F^{\mathrm{ext}}_{{\text{$r$-spin}}}((-r)t^1,\ldots,(-r)t^N,(-r)s).$$ Clearly, it satisfies the open WDVV equations, associated to $F_{A_N}$, together with condition . In [@BCT18 Proposition 5.1] the authors found an explicit formula for the coefficients of $F^{\mathrm{ext}}_{{\text{$r$-spin}}}$ that gives $$\begin{gathered}
{\left<}\tau_{\alpha_1}\ldots\tau_{\alpha_n}\sigma^k{\right>}^o_{A_N}=
\begin{cases}
(n+k-2)!,&\text{if $\sum_{i=1}^n(N+2-\alpha_i)+k=N+2$},\\
0,&\text{otherwise},
\end{cases}\end{gathered}$$ where we use the notation $${\left<}\tau_{\alpha_1}\ldots\tau_{\alpha_n}\sigma^k{\right>}^o_{A_N}:=\left.\frac{\d^{n+k} F^o_{A_N}}{\d t^{\alpha_1}\ldots\d t^{\alpha_n}\d s^k}\right|_{t^*=s=0}.$$ In [@Bur18] the author proved that the coefficients of the function $F^o_{A_N}$ are related to the expression of the coordinates $v_k$ in the terms of the flat coordinates $t^1,\ldots,t^N$ on the Saito Frobenius manifold of the singularity $A_N$ by $$\begin{gathered}
\label{eq:FoAN and the flat coordinates}
\frac{\d F^o_{A_N}}{\d s}=\frac{s^{N+1}}{N+1}+\sum_{k=1}^Ns^{k-1}v_k(t^1,\ldots,t^N).\end{gathered}$$

Generalized Dubrovin–Saito construction for the singularities $A$ and $D$ {#section: Mwext explicit}
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In this section, for the singularites of types $A$ and $D$ we present a generalization of the Dubrovin–Saito construction that 