We make a systematic investigation of stationary cylindrically symmetric solutions to the five-dimensional Einstein and Einstein-Gauss-Bonnet equations. Apart from the fivedimensional neutral cosmic string metric, we find two new exact solutions which qualify as cosmic strings, one corresponding to an electrically charged cosmic string, the other to an extended superconducting cosmic string surrounding a charged core. In both cases, test particles are deflected away from the singular line source. We extend both kinds of solution to exact multi-cosmic-string solutions.