Trajectories in interlaced integral pencils of 3-dimensional analytic vector fields are o-minimal

Let $\xi$ be an analytic vector field at $({\mathbb{R}}^3,0)$ and $\mathcal{I}$ be an analytically non-oscillatory integral pencil of $\xi$ ; i.e., $\mathcal{I}$ is a maximal family of analytically non-oscillatory trajectories of $\xi$ at 0 all sharing the same iterated tangents. We prove that if $\mathcal{I}$ is interlaced, then for any trajectory $\mathrm{\Gamma }\in \mathcal{I}$ , the expansion ${\mathbb{R}}_{\mathrm{a}\mathrm{n},\mathrm{\Gamma }}$ of the structure ${\mathbb{R}}_{\mathrm{a}\mathrm{n}}$ by $\mathrm{\Gamma }$ is model-complete, o-minimal and polynomially bounded.