A ( 2 + 1 ) - dimensional integrable spin model ( the M - XXII equation ) and Differential geometry of curves / surfaces

Using the differential geometry of curves and surfaces the Lakshmanan equivalent counterpart of the M-XXII equation is found. It is shown that these equations are too gauge eqivalent to each other. Also the gauge equivalence between the Strachan and M-XXII q equations is established. Some integrals of motion are presented. It is well known that the spectral parameter of the some (2+1)-dimensional soliton equations satisfies the following equations: λ t = κλ n λ y (a nonisospectral problems). We present the simplest exact solutions of this equation.