  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 $y = (y_1,y_2)$. 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.

Dynamical Systems;Classical Analysis and ODEs;Differential Geometry;Logic