that produces solutions of the open WDVV equations. In our paper we present such a generalization for the singularities of types $A$ and $D$. For the $A$-singularity our construction gives a polynomial solution that coincides with the one from [@BCT18; @BCT19]. For the $D$-singularity our solution has a simple pole along the variable $s$.

Additionally, in both $A$- and $D$-cases our solution of the open WDVV equations has the following remarkable feature. The Saito Frobenius manifold of a simple singularity has two natural coordinate systems. The first one is given by the parameters of a universal unfolding of a simple singularity. The second coordinate system is given by the flat coordinates of the metric. We show that for the singularities $A$ and $D$ the transition functions between these two coordinate systems coincide with the coefficients of powers of the variable $s$ in the expansion of our solution of the open WDVV equations.

The Saito Frobenius manifolds of simple singularities together with their certain submanifolds form a class of Frobenius manifolds, that is, via a construction of B. Dubrovin [@Dub98], in a natural bijection with the class of finite irreducible Coxeter groups (see also [@Zub94]). This class of Frobenius manifolds plays 