Classical solutions of the Gravitating Abelian Higgs Model

We consider the classical equations of the gravitating Abelian-Higgs model in an axially symmetric ansatz. More properties of the solutions of these equations (the Melvin and the sting branches) are presented. These solutions are also constructed for winding numbers N=2. It is shown that these vortices exist in attractive and repulsive phases, separated by the value of the Higgs coupling constant parameter leading to self-dual equations.