produces a flat F-manifold that extends the Saito Frobenius manifold and, therefore, gives a solution of the open WDVV equations. In the $A_N$-case this solution coincides with the function $F^o_{A_N}$. In both $A$- and $D$-cases, the coefficients of powers of the variable $s$ in this solution coincide with the transition functions between two coordinate systems on the Saito Frobenius manifold.

$A_N$-case
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Consider the space $M^{\mathrm{ext}}_{A_N}:={\mathbb C}^{N+1}$ with coordinates $v_1,\ldots,v_{N+1}$. Consider the quotient ring $${{\widehat{\mathcal{A}}}}^{{\mathrm{ext}}}_{A_N}:={{\mathbb{C}}}[x,y,w,v_1,\ldots,v_{N+1}] \Big/ \left(w - {{\partial}}_x\Lambda_{A_N}, {{\partial}}_y \Lambda_{A_N}, wx - v_{N+1} w \right).$$ As a ${\mathbb C}[v_1,\ldots,v_{N+1}]$-module, the space ${{\widehat{\mathcal{A}}}}^{{\mathrm{ext}}}_{A_N}$ is free of dimension $N+1$, and the elements $[1],[x],\ldots,[x^{N-1}],[w]$ form a basis. To show that any other element $[x^i y^j w^k]$ can be expressed in terms of them, first note that, obviously, $[y]=0$ and $[w x]=v_{N+1}[w]$. We also see that $$\begin{gathered}
\label{eq:w2 for AN}
[w^2]=\left[w\left(x^N+\sum_{k=2}^{N}(k-1)v_k x^{k-2}\right)\right]=\left(v_{N+1}^N+\sum_{k=2}^N(k-1)v_k v_{N+1}^{k-2}\right)[w].\end{gathered}$$ Using the relation $[x^N]=[w]-\sum_{k=2}^N(k-1)v_k[x^{k-2}]$, we can express any element $[x^p]$ with $p\ge N$ in terms of the elements $[1],[x],\ldots,[x^{N-1}],[w]$.

Identifying the ${\mathbb C}[v_1,\ldots,v_{N+1}]$-modules ${\mathcal{T}}^{\mathrm{alg}}_{M^{\mathrm{ext}}_{A_N}}(M^{\mathrm{ext}}_{A_N})$ and ${{\widehat{\mathcal{A}}}}^{\mathrm{ext}}_{A_N}$ via the isomorphism $\Psi^{\mathrm{ext}}_{A_N}$ defined by $$\begin{gathered}
\Psi_W^{\mathrm{ext}}\left(\frac{{{\partial}}}{{{\partial}}v_k}\right) := [x^{k-1}], \quad 1 \le k \le N,\qquad \Psi_W^{\mathrm{ext}}\left(\frac{{{\partial}}}{{{\partial}}v_{N+1}} \right) := [w],\end{gathered}$$ by Remark \[remark:algebraic construction of multiplication\], we endow the tangent spaces $T_pM_{A_N}^{\mathrm{ext}}$ with a 