  We examine the Standard Model field configurations near cosmic strings in a
particular class of models. This class is defined by the condition that the
generator of the flux in the string, $T_s$, commutes with the Standard Model
Lie algebra. We find that if the Standard Model Higgs carries a charge $F_h /2$
under $T_s$, cosmic string solutions have Z-flux $\Phi_Z =[n-F_h
N/F_{\phi}]4\pi \cos \theta_w /g$, where $n$ is any integer and $4\pi
N/qF_{\phi}$ is the flux of the gauge field associated with $T_s$. Only the
configuration with the smallest value of $|n-F_h N/F_{\phi}|$ is stable,
however. We argue that the instabilities found at higher $\Phi_Z$ are just
associated with paths in configuration space reducing $|n-F_h N/F_{\phi}|$ by
one unit. This contradicts recent claims that the instabilities in such models
represent the spontaneous generation of current along the string. We also show
that the stable strings have no Standard Model fermion zero modes: therefore
there is no possibility of supercurrents carried by Standard Model particles in
this class of models.
