We give an overview of our earlier classification results in [DW4] and [DW6] for superpotentials of scalar curvature type of the cohomogeneity one Ricci-flat equations. We then give an account of the classification in the case where the isotropy repr
esentation of the principal orbit consists of exactly three distinct irreducible real summands--the leftover case from [DW6].
We extend our previous classification of superpotentials of ``scalar curvature type for the cohomogeneity one Ricci-flat equations. We now consider the case not covered in our previous paper, i.e., when some weight vector of the superpotential lies o
utside (a scaled translate of) the convex hull of the weight vectors associated with the scalar curvature function of the principal orbit. In this situation we show that either the isotropy representation has at most 3 irreducible summands or the first order subsystem associated to the superpotential is of the same form as the Calabi-Yau condition for submersion type metrics on complex line bundles over a Fano Kahler-Einstein product.
