Lie-Backlund vector fields for the nonlinear system,Q t=AQxx+F(Qx,Q)

We have analyzed the class of nonlinear second-order equations written asQ_t=AQ_xx +F(Qx, Q) withQ =(_v^u) andA, F are, respectively, matrix and vector functions depending onQ, Q_x, from the point of view of Lie-Backlund vector fields. When the vector functionF does not depend onQ_x, these equation set reduces to the coupled diffusion equations discussed by Steeb. But our generalized system encompasses a large class of physically meaning full nonlinear equations, such as (i) dispersive water waves and (ii) a completely anisotropic Heisenberg spin chain. We also exhibit a new nonlinear coupled system which do have nontrivial Lie-Backlund vector fields. Also our approach yields more information about the symmetry generators for a wider class of nonlinear equations than the function space approach of Fuchsteiner in a much simpler way.