  We investigate the conformal invariant Lagrangian of the self-gravitating
U(1) scalar-gauge field and find new features of the model on the
time-dependent axially symmetric Bondi-Marder spacetime. By considering the
conformal symmetry as exact at the level of the Lagrangian and broken in the
vacuum, a consistent model is found with an exact solution of the vacuum
Bondi-Marder spacetime $g_{\mu\nu}=\omega^2 \bar g_{\mu\nu}$, where $\omega$ is
the conformal factor and $\bar g_{\mu\nu}$ the `un-physical` spacetime. If we
try to match this vacuum solution onto the interior vortex solution of the
coupled Einstein-scalar-gauge field, we need, besides the matching conditions,
constraint equations in order to obtain a topological regular description of
the small-scale behaviour of the model. Probably, one needs the
five-dimensional warped counterpart model, where the 5D dilaton field act as a
warp factor. Moreover, the tracelessness of the energy-momentum tensor could
then be maintained by a contribution from the bulk.
