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I searched on the internet superficially but I couldn't find it.
Is there any reference that find the solution to Einstein's Field Equations for a rotating Cosmic String?
Personally haven't got the time to do the computation for the rotating flux tube extension of a 2d vortex solution to the Higgsed-$U(1)$ Yang Mills, but that's where I would definitely start from.
The "Catalogue of Spacetimes" review of exact solutions to Einstein's equation, Mueller & Graves (2009), is the first place I would look. And sure enough section 2.22 might be what you are looking for. They pulled that metric from Perlick (2004), an LRR on gravitational lensing, which can point you further down the reference rabbit hole.