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In this paper, we study integrable multilayer spin systems, namely, themultilayer M-LIII equation. We investigate their relation with the geometricflows of interacting curves and surfaces in some space $R^{n}$. Then we presenttheir the Lakshmanan equivalent counterparts. We show that these equivalentcounterparts are, in fact, the vector nonlinear Schr\"odinger equation (NLSE).It is well-known that the vector NLSE is equivalent to the $\Gamma$-spinsystem. Also, we have presented the transformations which give the relationbetween solutions of the $\Gamma$-spin system and the multilayer M-LIIIequation. It is interesting to note that the integrable multilayer M-LIIIequation contains constant magnetic field ${\bf H}$. It seems that thisconstant magnetic vector plays an important role in theory of "integrablemultilayer spin system" and in nonlinear dynamics of magnetic systems. Finally,we present some classes of integrable models of interacting vortices.

Integrable geometric flows of interacting curves/surfaces, multilayer spin systems and the vector nonlinear Schr\"odinger equation