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Results are presented from a spectroscopic study of the Ae/Be Herbig star HD 259431 conducted from 2010 to 2019 using high-resolution spectrographs at: (a) the Crimean Astrophysical Observatory, (b) European Southern Observatory ESO (Chile), (c) the OAN SPM Observatory in Mexico, and (d) the Kourovskaya Astronomical Observatory at the Urals Federal University. 245 spectra of the object were obtained in the neighborhood of circumstellar lines (mainly, Hα, Hβ, HeI 5876, FeII 4923, and DNaI), as well as some lines from its atmosphere (HeI 4009, 4026). This article is concerned only with analyzing the cyclical variability of the parameters of some circumstellar lines on a time scale of the order of one day. Antiphase variations in the bisector velocity of the emission profile of Hβ and the velocity of the boundary of the red wing of the HeI 5876 absorption line is observed. A period P = 2.839 days is found in these variations. Assuming that this value may be the period of rotation of the star, an angle of inclination of the axis of rotation to the line of sight of i = 48° ± 7° is estimated. Since at different phases of this period the profiles of Hβ and Hγ looked like either a PCyg-profile or a double emission with a depression on the red side, it was suggested that the observed variability is associated with the existence of a magnetosphere around HD 259431, which has an inclination of the magnetic axis relative to the axis of rotation. In this case over one rotation of the star, along the line of sight an accretion and then an outflow flux of matter occurs, since the regions with these two types of kinematics according to the model lie at different magnetic latitudes. The existence of the magnetosphere is supported by the very high velocities of the boundary of the red wing of the HeI 5876 absorption line (up to +400 km/s), observed at the phases when an accretion flow is on the line of sight. It was found that similar variations are also manifested by the Hδ, HeI 6678, and OI 7773 lines. At the same time, in a number of lines formed in the circumstellar medium, far from the region of the disk/star interaction region, this kind of cyclical variability was not observed. We note that this is the first case of detecting a magnetosphere in a Be Herbig star of such an early class (B6). Earlier it was assumed that magnetospheres cannot be observed in Be Herbig stars. Keywords: Ae/Be Herbig stars: circumstellar shells: disk accretion: wind: HD 259431 [ABSTRACT FROM AUTHOR]