the non-Heisenberg structure of the spin-spin interaction affects even the traditional phases. In recent years, there have been intensive studies of unconventional orderings in magnets with $S\geqslant 1$ [@Toth; @BH; @NC; @Fridman; @Kosm; @Kolezhuk; @Kov1].

Nowadays, the interest in high-spin systems is attracted by the studies of ultracold atomic gases providing remarkable opportunities to examine and model various effects and phenomena in quantum many-body systems in a well controlled manner. In particular, quantum gases loaded in an optical lattice represent an artificial but effective simulator of magnetic phenomena in crystalline systems [@Sotnikov1; @Sotnikov2]. The first theoretical studies of magnetic phases and corresponding excitations in dilute Bose gases with condensate [@Ohmi; @APS; @Ho] were stimulated by experiments on optical trapping a condensate of ${}^{23}$Na spin-1 atoms [@Stamper]. A comprehensive study of the so-called spinor Bose gases, including those of spin-2 and spin-3 atoms, has been reviewed in Refs. [@Ueda; @Kurn], where the interaction between the atomic degrees of freedom was taken as a bilinear form in spin operators, like in the usual Heisenberg model. However, as was mentioned, such a form of interaction energy is not sufficient to describe properly the magnetic properties of high-spin systems.

In this paper, proceeding solely 