  The static cylindrically symmetric solutions of the gravitating Abelian Higgs
model form a two parameter family. In this paper we give a complete
classification of the string-like solutions of this system. We show that the
parameter plane is composed of two different regions with the following
characteristics: One region contains the standard asymptotically conic cosmic
string solutions together with a second kind of solutions with Melvin-like
asymptotic behavior. The other region contains two types of solutions with
bounded radial extension. The border between the two regions is the curve of
maximal angular deficit of $2\pi$.
