Solutions of definable ODEs with regular separation and dichotomy interlacement versus Hardy

We introduce a notion of regular separation for solutions systems ODEs $y'=F(x,y)$, where $F$ is definable in polynomially bounded o-minimal structure and $y=(y\_1,y\_2)$. Given pair with flat contact, we prove that, if one them has the property separation, either interlaced or generates Hardy field. adapt this result to trajectories three-dimensional vector fields coefficients. In particular case real analytic fields, it improves dichotomy interlaced/separated certain integral pencils, obtained by F. Cano, R. Moussu third author. context, show that set asymptotic formal invariant curve never empty represented subanalytic minimal dimension containing curve. Finally, how construct examples curves which are transcendental respect sets, using so-called (SAT) property, introduced J.-P. Rolin, Shaefke