On the self-similar Jordan arcs admitting structure parametrization

We study the attractors $\gamma$ of a finite system $\mathscr{S}$ of contraction similarities $S_j$ $(j=1,\dots,m)$ in $\mathbb{R}^d$ which are Jordan arcs. We prove that if a system $\mathscr{S}$ possesses a structure parametrization $(\mathscr{T},\varphi)$ and $\mathscr{F}(\mathscr{T})$ is the associated family of $\mathscr{T}$ then we have one of the following cases:
1. The identity mapping $\operatorname{Id}$ does not belong to the closure of $\mathscr{F}(\mathscr{T})$. Then $\mathscr{S}$ (if properly rearranged) is a Jordan zipper.
2. The identity mapping $\operatorname{Id}$ is a limit point of $\mathscr{F}(\mathscr{T})$. Then the arc $\gamma$ is a straight line segment.
3. The identity mapping $\operatorname{Id}$ is an isolated point of $\overline{\mathscr{F}(\mathscr{T})}$.
We construct an example of a self-similar Jordan curve which implements the third case.