  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.
