In that paper, firstly we get two additional different non-null space curve evolutions in Minkowski 3-space by considering Landau-Lifshitz (LL-) equation where we identify the spin vector with the binormal vector and the normal vector of these curves, respectively. Then, we obtain some links for constructing the moving non-null space curves by using the integrable LL- equation. Finally, we give as an application, the exact solution of the moving non-null curve evolutions obtained by taking the spin vector the normal vector of the curve and we showed graphically that these solutions are wave solutions.