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Spin nematic state for a spin S=3/2 isotropic non-Heisenberg magnet

$S=3/2$ system with general isotropic nearest-neighbor exchange within a mean-field approximation possesses a magnetically ordered ferromagnetic state and antiferromagnetic state, and two different spin nematic states, with zero spin expectation values. Both spin nematic phases display complicated symmetry break, including standard rotational break described by the vector-director $\vec {u}$ and specific symmetry break with respect to the time reversal. The break of time reversal is determined by non-trivial quantum averages cubic over the spin components and can be described by unit "pseudospin" vector $\vec σ$. The vectors $\vec σ$ on different sites are parallel for a nematic state, and $\vec σ$'s are antiparallel for different sublattices for an antinematic phase.