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Motion of curves and surfaces in $\R^3$ lead to nonlinear evolution equationswhich are often integrable. They are also intimately connected to the dynamicsof spin chains in the continuum limit and integrable soliton systems throughgeometric and gauge symmetric connections/equivalence. Here we point out thefact that a more general situation in which the curves evolve in the presenceof additional self consistent vector potentials can lead to interestinggeneralized spin systems with self consistent potentials or soliton equationswith self consistent potentials. We obtain the general form of the evolutionequations of underlying curves and report specific examples of generalized spinchains and soliton equations. These include principal chiral model and variousMyrzakulov spin equations in (1+1) dimensions and their geometricallyequivalent generalized nonlinear Schr\"odinger (NLS) family of equations,including Hirota-Maxwell-Bloch equations, all in the presence of selfconsistent potential fields. The associated gauge equivalent Lax pairs are alsopresented to confirm their integrability.

Integrable Motion of Curves in Self-Consistent Potentials : Relation to Spin Systems and Soliton Equations