  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 vs separated of certain integral pencils, as studied in [CMS04]. 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 from [RSS07].
