  We continue the classification of 2-dimensional scale-invariant rigid special
Kahler (RSK) geometries. This classification was begun in [hep-th/0504070]
where singularities corresponding to curves of the form y^2=x^6 with a fixed
canonical basis of holomorphic one forms were analyzed. Here we perform the
analysis for the y^2=x^5 type singularities. (The final maximal singularity
type, y^2=x^3(x-1)^3, will be analyzed in a later paper.) These singularities
potentially describe the Coulomb branches of N=2 supersymmetric field theories
in four dimensions. We show that there are only 13 solutions satisfying the
integrability condition (enforcing the RSK geometry of the Coulomb branch) and
the Z-consistency condition (requiring massless charged states at
singularities). Of these solutions, one has a marginal deformation, and
corresponds to the known solution for certain Sp(2) gauge theories, while the
rest correspond to isolated strongly interacting conformal field theories.
