Conclusions
There are Static, Stable Topological Defect solutions even in theories where the homotopy groups of the vacuum are trivial. These are the Embedded Defects.
The simplest paradigm of an Embedded Defect is the Semilocal Vortex. It has well defined stability parameter sector.
The Semilocal Vortex exists also as a Stable Solution in the Standard Electroweak Model of Particle Physics called Electroweak Vortex.
The Electroweak Vortex is Unstable for the physically realized values of parameters of the Standard Electroweak Model.
The Semilocal Vortex becomes stable for all parameter values when the SU(2) symmetry gets explicitly broken. Then non-trivial core structure develops for a certain parameter range.