Integrable spin systems are an important subclass of integrable (soliton) nonlinear equations. They play important role in physics and mathematics. At present, many integrable spin systems were found and studied. They are related with the motion of 3-dimensional curves. In this paper, we consider a model of two moving interacting curves. Next, we find its integrable reduction related with some integrable coupled spin system. Then we show that this integrable coupled spin system is equivalent to the famous Manakov system.

In this paper, we study integrable multilayer spin systems, namely, the multilayer M-LIII equation. We investigate their relation with the geometric flows of interacting curves and surfaces in some space R n . Then we present their the Lakshmanan equivalent counterparts. We show that these equivalent counterparts are, in fact, the vector nonlinear Schrödinger equation (NLSE). It is well-known that the vector NLSE is equivalent to the Γ-spin system. Also, we have presented the transformations which give the relation between solutions of the Γ-spin system and the multilayer M-LIII equation. It is interesting to note that the integrable multilayer M-LIII equation contains constant magnetic field H. It seems that this constant magnetic vector plays an important role in theory of "integrable multilayer spin system" and in nonlinear dynamics of magnetic systems. Finally, we present some classes of integrable models of interacting vortices. *

In recent years, symmetry in abstract partial differential equations has found wide application in the field of nonlinear integrable equations. The symmetries of the corresponding transformation groups for such equations make it possible to significantly simplify the procedure for establishing equivalence between nonlinear integrable equations from different areas of physics, which in turn open up opportunities to easily find their solutions. In this paper, we study the symmetry between differential geometry of surfaces/curves and some integrable generalized spin systems. In particular, we investigate the gauge and geometrical equivalence between the local/nonlocal nonlinear Schrödinger type equations (NLSE) and the extended continuous Heisenberg ferromagnet equation (HFE) to investigate how nonlocality properties of one system are inherited by the other. First, we consider the space curves induced by the nonlinear Schrödinger-type equations and its equivalent spin systems. Such space curves are governed by the Serret–Frenet equation (SFE) for three basis vectors. We also show that the equation for the third of the basis vectors coincides with the well-known integrable HFE and its generalization. Two other equations for the remaining two vectors give new integrable spin systems. Finally, we investigated the relation between the differential geometry of surfaces and integrable spin systems for the three basis vectors.

Among nonlinear evolutionary equations integrable ones are of particular interest since only in this we case can theoretically study the model in detail and in-depth. In the present, we establish the geometric connection of the well-known integrable two-component Manakov system with a new two-layer spin system. This indicates that the latter system is also integrable. In this formalism, geometric invariants define some important conserved quantities associated with two interacting curves, and with the corresponding nonlinear evolution equations.

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