symmetry.

In the same way, as the WDVV equations appeared in Gromov–Witten theory, another system of non-linear PDEs, called the *open WDVV equations*, appeared more recently in open Gromov–Witten theory [@HS12 Theorem 2.7] (see also [@PST14; @BCT18; @BCT19]). Let $F = F(t^1,\dots,t^N)$ be a solution of the WDVV equations , satisfying condition . The open WDVV equations associated to $F$ are the following PDEs for a function $F^o = F^o(t^1,\ldots,t^N,s)$, depending on an additional variable $s$: $$\begin{aligned}
\frac{\d^3F}{\d t^\alpha\d t^\beta\d t^\mu}\eta^{\mu\nu}\frac{\d^2F^o}{\d t^\nu\d t^\gamma}+\frac{\d^2F^o}{\d t^\alpha\d t^\beta}\frac{\d^2F^o}{\d s\d t^\gamma}=&\frac{\d^3F}{\d t^\gamma\d t^\beta\d t^\mu}\eta^{\mu\nu}\frac{\d^2F^o}{\d t^\nu\d t^\alpha}+\frac{\d^2F^o}{\d t^\gamma\d t^\beta}\frac{\d^2F^o}{\d s\d t^\alpha},&&1\le\alpha,\beta,\gamma\le N,\label{eq:open WDVV,1}\\
\frac{\d^3F}{\d t^\alpha\d t^\beta\d t^\mu}\eta^{\mu\nu}\frac{\d^2F^o}{\d t^\nu\d s}+\frac{\d^2F^o}{\d t^\alpha\d t^\beta}\frac{\d^2F^o}{\d s^2}=&\frac{\d^2F^o}{\d s\d t^\beta}\frac{\d^2F^o}{\d s\d t^\alpha},&&1\le\alpha,\beta\le N.\label{eq:open WDVV,2}\end{aligned}$$ Solutions of equations , , relevant in open Gromov-Witten theory and also in the works [@PST14; @BCT18; @BCT19], satisfy the additional condition $$\begin{gathered}
\label{eq:unit condition for Fo}
\frac{\d^2 F^o}{\d t^1\d t^\alpha}=0,\qquad \frac{\d^2 F^o}{\d t^1\d s}=1.\end{gathered}$$

The solutions of the open WDVV equations from the works [@BCT18; @BCT19] are associated to the Saito Frobenius manifold of the $A$-singularity and they were constructed using ideas of FJRW theory. So it is natural to ask whether the Dubrovin–Saito construction of the Frobenius manifolds corresponding to simple singularities admits a generalization, 