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The spin-resolved electronic band structure of Au-induced metallic atomic wires on a vicinal silicon surface, Si(553), was investigated using spin- and angle-resolved photoelectron spectroscopy. We directly measured the spin polarization of three partially filled one-dimensional metallic bands, a one-third-filled band, and the doublet of nearly half-filled bands. For the half-filled doublet, the strong apparent spin polarization was observed near the Fermi energy with a minor out-of-plane spin component. This observation is consistent with the Rashba-type spin-orbit splitting and with a recent experiment on a similar doublet of Si(557)-Au. In contrast, the one-third-filled band does not show a substantial spin polarization within the experimental accuracy, indicating a much smaller spin splitting, if any. These results are discussed for the origin of the partially filled bands and for the intriguing broken-symmetry ground state observed at low temperature.