The Darboux-Backlund transformation without using a matrix representation

We present a new method to construct soliton solutions for a large class of nonlinear systems which are associated with matrix spectral problems of hi gh degree (including a reduction of the Lame equations and a generalized si ne-Gordon system). The spectral problems are represented in terms of Cliffo rd numbers. Then the solutions can be obtained in a straightforward way wit hout using any matrix representation. Among the spectral problems under con sideration is the Dirac equation for a massless particle in the electromagn etic field and its generalizations.