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We analyze spin-up/spin-down of the neutron star in Be X-ray binary systemGX\,304-1 observed by \textit{Swift}/XRT and \textit{Fermi}/GBM instruments inthe period of the source activity from April 2010 to January 2013 and discusspossible mechanisms of angular momentum transfer to/from the neutron star. Weargue that the neutron star spin-down at quiescent states of the source with anX-ray luminosity of $L_x\sim 10^{34}$~erg s$^{-1}$ between a series of Type Ioutbursts and spin-up during the outbursts can be explained by quasi-sphericalsettling accretion onto the neutron star. The outbursts occur near the neutronstar periastron passages where the density is enhanced due to the presence ofan equatorial Be-disc tilted to the orbital plane. We also propose anexplanation to the counterintuitive smaller spin-up rate observed at higherluminosity in a double-peak Type I outburst due to lower value of the specificangular momentum of matter captured from the quasi-spherical wind from theBe-star by the neutron star moving in an elliptical orbit with eccentricity$e\gtrsim 0.5$.

Spin-up/spin-down of neutron star in Be-X-ray binary system GX 304-1