It is generally assumed that through the Kibble mechanism topological defects persisted since the early universe. In this talk we discuss a model beyond the Standard Model that permits a non-standard type of cosmic strings. In order to study cosmic string solutions, we first promote the global symmetry U$(1)_{B-L}$ (baryon minus lepton number) to a local symmetry and add a new gauge coupling. The cancellation of gauge anomalies is achieved by adding a right-handed neutrino to each lepton generation. Moreover, a new Higgs field is added in order to give mass to the right-handed neutrino.
Finally, we study the field equations of motion of the two Higgs fields and the gauge field, in order to obtain the profiles of the cosmic strings. We compute these profiles numerically and discuss their physical meaning. Particularly we found a type of solutions that we call coaxial cosmic strings. In addition, we compute the string tension which is of the order of $10^{19}\,\text{GeV}^2$ and its gravitational coupling $\sim 10^{-30}$ which is below the constraint obtained by the LIGO collaboration.