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In this paper, we have studied the geometrical formulation of theLandau-Lifshitz equation (LLE) and established its geometrical equivalentcounterpart as some generalized nonlinear Schr\"{o}dinger equation. When theanisotropy vanishes, from this result follows the well-known resultscorresponding for the isotropic case, i.e. to the Heisenberg ferromagnetequation and the focusing nonlinear Schr\"{o}dinger equation. The relationsbetween the LLE and the differential geometry of space curves in the local andnonlocal cases are studied. Using the well-known Sym-Tafel formula, the solitonsurfaces induced by the LLE are briefly considered.

Integrable motion of anisotropic space curves and surfaces induced by the Landau-Lifshitz equation