Einstein-Bogomolnyi metrics, which physically models the Cosmic Strings, solve the Einstein’s Fields Equation coupled with an Abelian gauge field and a Higgs field. In this talk, I will present a general existence and uniqueness theorem for Einstein-Bogomolnyi metrics on Riemann sphere. I will also discuss the behaviors of the metrics as the volume approaches the lower bound and infinity respectively, and the moduli space problem. Part of this talk is based on the joint work with Luis-Alvarez, Garcia-Fernandez, Garcia-Prada and Pingali.