In this paper, we have studied the geometrical formulation of the Landau-Lifshitz equation (LLE) and established its geometrical equivalent counterpart as some generalized nonlinear Schrödinger equation. When the anisotropy vanishes, from this result we obtain the well-known results corresponding for the isotropic case, i.e. to the Heisenberg ferromagnet equation and the focusing nonlinear Schrödinger equation. The relations between the LLE and the differential geometry of space curves in the local and in nonlocal cases are studied. Using the well-known Sym-Tafel formula, the soliton surfaces induced by the LLE are briefly considered.

In this paper, we study the Kuralay equations, namely, the Kuralay-I equation (K-IE) and the Kuralay-II equation (K-IIE). The integrable motion of space curves induced by these equations is investigated. The gauge equivalence between these two equations is established. With the help of the Hirota bilinear method, the simplest soliton solutions are also presented. The nonlocal and dispersionless versions of the Kuralay equations are considered.