  Sufficiently differentiable oval billiards always have invariant rotational
curves, but there are only two types of ovals with an invariant horizontal
circle in its phase-space: the constant width ovals and some very special
symmetric curves.
  In this work we study the dynamics near the horizontal circle for the
billiard map of those two types of ovals, and show that the horizontal circle
is approached, from both sides, by other invariant rotational curves. We will
also describe some dynamical consequences.
