Let ξ be an analytic vector field at (R 3 , 0) and I be an analytically non-oscillatory integral pencil of ξ; i.e., I is a maximal family of analytically non-oscillatory trajectories of ξ at 0 all sharing the same iterated tangents. We prove that if I is interlaced, then for any trajectory Γ ∈ I, the expansion R an,Γ of the structure R an by Γ is model-complete, o-minimal and polynomially bounded.