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It has been shown that 5-dimensional general relativity action extended byappropriate quadratic terms admits a singular superconducting cosmic stringsolution. We search for cosmic strings endowed with similar and extendedphysical properties by directly integrating the non-linear matrix fieldequations thus avoiding the perturbative approach by which we constructed theabove-mentioned \textsl{exact} solution. The most general superconductingcosmic string, subject to some constraints, will be derived and shown to bemathematically \textsl{unique} up to linear coordinate transformations mixingits Killing vectors. The most general solution, however, is not globallyequivalent to the old one due to the existence of Killing vectors with closedorbits.

Quadratic superconducting cosmic strings revisited