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This paper has two aims. The first one is the construction problem ofalgebraic potentials of Frobenius manifolds. We show examples of suchpotentials for the cases of reflection groups of types $H_4,E_6,E_7,E_8$ andalso include those which are already known. The second one is an application ofsuch potentials to singularity theory. We introduce families of hypersurfacesof ${\bf C}^3$ which are deformations of $E_n$-singularities $(n=6,7,8)$ butare not the versal families of $E_n$-singularities. We study the properties ofthe families. In particular we show the correspondence between such familiesand the algebraic potentials constructed in the first aim. Moreover we discussthe relationship between the complex reflection groups $ST33$ and $ST34$ andthe two families corresponding to the $E_6$-singularity and the$E_7$-singularity.

The Construction Problem of Algebraic Potentials and Reflection Groups