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.