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00925njm 22002777a 4500 dc Le Gal, Olivier author Matusinski, Mickaël author Sanz Sánchez, Fernando author 2021 We introduce a notion of regular separation for solutions of systems of ODEs y'=F(x; y), where F is definable in a polynomially bounded o-minimal structure and y=(y1,y2). Given a pair of solutions with flat contact, we prove that, if one of them has the property of regular separation, the pair is either interlaced or generates a Hardy field. We adapt this result to trajectories of three-dimensional vector fields with definable coefficients. In the particular case of real analytic vector fields, it improves the dichotomy interlaced/separated of certain integral pencils, obtained by F. Cano, R. Moussu and the third author. In this context, we show that the set of trajectories with the regular separation property and asymptotic to a formal invariant curve is never empty and it is represented by a subanalytic set of minimal dimension containing the curve. Finally, we show how to construct examples of formal invariant curves which are transcendental with respect to subanalytic sets, using the so-called (SAT) property, introduced by J.-P. Rolin, R. Shaefke and the third author. Rev. Mat. Iberoam. 38 (2022), no. 5, 1501–1527 0213-2230 https://uvadoc.uva.es/handle/10324/68129 10.4171/RMI/1311 1501 5 1527 Revista Matemática Iberoamericana 38 2235-0616 Solutions of definable ODEs with regular separation and dichotomy interlacement versus Hardy