equations for the $A$-singularity {#section:extended r-spin}
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Here we explain how the Saito Frobenius manifold of the $A$-singularity together with a certain solution of the open WDVV equations appear in the intersection theory on the moduli spaces of algebraic curves.

Let $r=N+1$. For integers $0\le\alpha_1,\ldots,\alpha_n\le r-1$, using the geometry of algebraic curves with an $r$-spin structure, one can construct a cohomology class $$W^r_{0,n}(\alpha_1,\ldots,\alpha_n)\in H^{2d}({\overline{\mathcal M}}_{0,n},\mbQ),\quad d=\frac{\sum\alpha_i-(r-2)}{r},$$ called [*Witten’s class*]{}, on the moduli space ${\overline{\mathcal M}}_{0,n}$ of stable curves of genus $0$ with $n$ marked points (see e.g. [@PPZ16]). Here we assume that the class $W^r_{0,n}(\alpha_1,\ldots,\alpha_n)$ is equal to zero, if $d$ is not an integer or $d<0$. The class $W^r_{0,n}(\alpha_1,\ldots,\alpha_n)$ vanishes, if one of the $\alpha_i$’s is equal to $r-1$. Consider the generating series $$F_{{\text{$r$-spin}}}(t^1,\ldots,t^{r-1}):=\sum_{n\ge 3}\frac{1}{n!}\sum_{0\le\alpha_1,\ldots,\alpha_n\le r-2}\left(\int_{{\overline{\mathcal M}}_{0,n}}W^r_{0,n}(\alpha_1,\ldots,\alpha_n)\right)\prod_{i=1}^n t^{\alpha_i+1}.$$ The functions $F_{A_N}$ and $F_{{\text{$r$-spin}}}$ are related by [@JKV01a] $$F_{A_N}(t^1,\ldots,t^N)=(-r)^{-3}F_{{\text{$r$-spin}}}((-r)t^1,\ldots,(-r)t^N).$$ This is one of the simplest cases of mirror symmetry. Denote $${\left<}\tau_{\alpha_1}\ldots\tau_{\alpha_n}{\right>}_{A_N}:=\left.\frac{\d^n F_{A_N}}{\d t^{\alpha_1}\ldots\d t^{\alpha_n}}\right|_{t^*=0},\quad 1\le\alpha_1,\ldots,\alpha_n\le N.$$ We have (see e.g. [@LVX17 page 4]) $$\begin{gathered}
\label{eq:3-point and 4-point for AN}
{\left<}\tau_{\alpha_1}\tau_{\alpha_2}\tau_{\alpha_3}{\right>}_{A_N}=\delta_{\alpha_1+\alpha_2+\alpha_3,N+2},\qquad{\left<}\tau_{\alpha_1}\tau_{\alpha_2}\tau_{\alpha_3}\tau_{\alpha_4}{\right>}_{A_N}=-\min(\alpha_i-1,N+1-\alpha_i).\end{gathered}$$ These formulas will be used later.

In [@JKV01b] the authors noticed that the construction of Witten’s class $W^r_{0,n}(\alpha_1,\ldots,\alpha_n)$ can be extended to the case when $\alpha_1=-1$ and $0\le\alpha_2,\ldots,\alpha_n\le r-1$. In 