Cosmic strings with cosmological constant

All static and cylindrically symmetric vacuum solutions of Einstein's field equation with cosmological constant Λ are found. We use these solutions to represent the exterior metric of a cosmic string. If Λ is negative, the exterior approaches the anti--de Sitter metric away from the string. If Λ is positive, the exterior metric is either an scrR 3 x scrS 1 universe with a curvature singularity, an scrR 2 x scrS 2 universe with a conical singularity, or an scrR 2 x scrS 2 universe with no singularity