  Electron motion in an atom driven by an intense linearly polarized laser
field can exhibit a laser-dressed stable state, referred to as the
Kramers-Henneberger (KH) state or KH atom. Up to now, the existence conditions
of this state rely on the presence of a double well in the KH potential,
obtained by averaging the motion over one period of the laser. However, the
approximation involved in the averaging is largely invalid in the region of the
double well structure; therefore this raises the question of its relevance for
identifying signatures of these exotic states. Here we present a method to
establish conditions for the existence of the KH atom based on a
nonperturbative approach. We show that the KH atom is structured by an
asymmetric periodic orbit with the same period as the laser field in a wide
range of laser parameters. Its imprint is clearly visible on the wavefunction
in quantum simulations. We identify the range of parameters for which this KH
state is effective, corresponding to an elliptic periodic orbit.
