  The static string-like solutions of the Abelian Higgs model coupled to
dilaton gravity are analyzed and compared to the non-dilatonic case. Except for
a special coupling between the Higgs Lagrangian and the dilaton, the solutions
are flux tubes that generate a non-asymptotically flat geometry. Any point in
parameter space corresponds to two branches of solutions with two different
asymptotic behaviors. Unlike the non-dilatonic case, where one branch is always
asymptotically conic, in the present case the asymptotic behavior changes
continuously along each branch.
