  We analyze the existence of string-like defects in a two-Higgs-doublet system
having $SU(2) \times U(1)_Y \times U(1)_{Y^{\prime}}$ as gauge group. We are
able to show that, when certain relations among the parameters hold, these
configurations satisfy a set of first order differential equations (Bogomol'nyi
equations) and their energy is proportional to their topological charge.}
