**5. Conclusions**

We analyzed the relativistic effect for hydrogenic Rydberg atoms or ions in a highfrequency linearly- or circularly-polarized laser field. For the general case, where the electron orbit is not perpendicular to the direction of the laser field, we showed that the precession of the electron orbit within its plane can vanish at some critical polar angle θc of the orbital plane. We calculated analytically the dependence of the critical angle on the angular momentum of the electron and on the parameters of the laser field.

For the particular situation, where the electron angular momentum is collinear with the linearly-polarized laser field, we demonstrated that the relativistic precession and the precession due to the laser field occur in the opposite directions. As a result, the combined effect of these two kinds of the precession is smaller than the absolute value of each of them. We showed that, by varying the ratio of the laser field strength *F* to the square of the laser field frequency ω, one can control the precession frequency of the electron orbit and even make the precession vanish, so that the elliptical orbit of the electron would become stationary. This is a counterintuitive result.

In general, any precession at some frequency Ω manifests in the radiation spectrum of a hydrogenic atom/ion as satellites separated from the unperturbed frequency of the spectral line by multiples of the precession frequency. So, the above situation where the elliptical orbit of the electron becomes stationary would manifest in the radiation spectrum as the disappearance of the satellites.

We hope that the fundamental nature of our analytical results for the hydrogenic atoms/ions, i.e., for atomic systems serving as the test bench of our understanding of atomic physics, makes the results significant.

**Author Contributions:** E.O.: conceptualization; investigation; writing—original draft preparation; writing—review and editing. N.K.: investigation; writing—original draft preparation; writing— review and editing. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work received no funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** All data is included in the manuscript.

**Conflicts of Interest:** The authors declare no conflict of interest.
