Multiplicity of Massive stars in the Milky Way (M3W). II. The brightest single O stars: variability and systemic velocities
(ABRIDGED) M3W studies the dynamical properties of massive stars in the solar vicinity, such as their systemic velocities, which are affected by variability effects: pulsations, line infilling by wind and disks, and incomplete knowledge about the central wavelengths. We want to increase the accuracy of the O-stars systemic velocities. We establish how different wavelength regions are affected by variability, determine the thresholds for the detection of spectroscopic binarity, and obtain accurate central wavelengths including the effects of contamination, forbidden lines, and wind infilling. We analyse 41 apparently single bright O stars. We used multi-epoch HR LiLiMaRlin spectroscopy cleaned with UNWIND and stellar models computed with FASTWIND. The epoch spectra were cross-correlated to study their variability. The RVs were computed from (a) cross-correlation of H and He lines with FASTWIND models, (b) profile fitting of metallic lines, and (c) cross-correlation of the full spectrum with a template star. A comparison between the RV dispersions for different wavelength regions reveals different groups of stars in terms of variability: (a) non-SG slow rotators, (b) fast rotators, (c) SGs, and (d) Oe stars. This is a manifestation of the different role of pulsations, winds, and disks in RV variability. We determine RV dispersion thresholds for the detection of new SBs. We also use a FASTWIND grid to establish the effects of (a) the presence of He II lines close to Balmer lines, (b) the existence of forbidden components in two He I lines, and (c) wind infilling in the determination of accurate RVs. With all of that, we determined RVs for our 41 stars, checked their accuracy, and discovered significant differences with literature values. Our analysis has further revealed four SB1 systems unknown in the summer of 2025, one new SB2, two new Oe stars, and three new Onfp stars.