LS 5039 is a high-mass gamma-ray binary hosting a compact object of unknown type. NuSTAR observed LS 5039 during its entire 3.9 day binary period. We performed a periodic signal search up to 1000 Hz which did not produce credible period candidates. We do see the 9.05 s period candidate, originally reported by Yoneda et al. 2020 using the same data, in the Fourier power spectrum, but we find that the statistical significance of this feature is too low to claim it as a real detection. We also did not find significant bursts or quasi-periodic variability. The modulation with the orbital period is clearly seen and remains unchanged over a decade long timescale when compared to the earlier Suzaku light curve. The joint analysis of the NuSTAR and Suzaku XIS data shows that the 0.7-70 keV spectrum can be satisfactory described by a single absorbed power-law model with no evidence of cutoff at higher energies. The slope of the spectrum anti-correlates with the flux during the binary orbit. Therefore, if LS 5039 hosts a young neutron star, its X-ray pulsations appear to be outshined by the intrabinary shock emission. The lack of spectral lines and/or an exponential cutoff at higher energies suggests that the putative neutron star is not actively accreting. Although a black hole scenario still remains a possibility, the lack of variability or Fe K$alpha$ lines, which typically accompany accretion, makes it less likely.
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