PSR B1259-63/LS 2883 is a gamma-ray binary system consisting of a pulsar in an eccentric orbit around a bright Oe stellar-type companion star that features a dense circumstellar disc. The high- and very-high-energy (HE, VHE) gamma-ray emission from PSR B1259-63/LS 2883 around the times of its periastron passage are characterised, in particular, at the time of the HE gamma-ray flares reported to have occurred in 2011, 2014, and 2017. Spectra and light curves were derived from observations conducted with the H.E.S.S.-II array in 2014 and 2017. A local double-peak profile with asymmetric peaks in the VHE light curve is measured, with a flux minimum at the time of periastron $t_p$ and two peaks coinciding with the times at which the neutron star crosses the companions circumstellar disc ($sim t_p pm 16$ d). A high VHE gamma-ray flux is also observed at the times of the HE gamma-ray flares ($sim t_p + 30$ d) and at phases before the first disc crossing ($sim t_p - 35$ d). PSR B1259-63/LS 2883 displays periodic flux variability at VHE gamma-rays without clear signatures of super-orbital modulation in the time span covered by H.E.S.S. observations. In contrast, the photon index of the measured power-law spectra remains unchanged within uncertainties for about 200 d around periastron. Lower limits on exponential cut-off energies up to $sim 40$ TeV are placed. At HE gamma-rays, PSR B1259-63/LS 2883 has now been detected also before and after periastron, close to the disc crossing times. Repetitive flares with distinct variability patterns are detected in this energy range. Such outbursts are not observed at VHEs, although a relatively high emission level is measured. The spectra obtained in both energy regimes displays a similar slope, although a common physical origin either in terms of a related particle population, emission mechanism, or emitter location is ruled out.
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