  The stability of an abelian (Nielsen-Olesen) vortex embedded in the
electroweak theory against W production is investigated in a gauge defined by
the condition of a single-component Higgs field. The model is characterized by
the parameters $\beta=({M_H\over M_Z})^2$ and $\gamma=\cos^2\theta_{\rm w}$
where $\theta_{\rm w}$ is the weak mixing angle. It is shown that the equations
for W's in the background of the Nielsen-Olesen vortex have no solutions in the
linear approximation. A necessary condition for the nonlinear equations to have
a solution in the region of parameter space where the abelian vortex is
classically unstable is that the W's be produced in a state of angular momentum
$m$ such that $0>m>-2n$. The integer $n$ is defined by the phase of the Higgs
field, $\exp(in\varphi)$. Solutions for a set of values of the parameters
$\beta$ and $\gamma$ in this region were obtained numerically for the case
$-m=n=1$. The possibility of existence of a stationary state for $n=1$ with W's
in the state $m=-1$ was investigated. The boundary conditions for the
Euler-Lagrange equations required to make the energy finite cannot be satisfied
at $r=0$. For these values of $n$ and $m$ the possibility of a finite-energy
stationary state defined in terms of distributions is discussed.
