Nonlinear (2+1)-dimensional field equations from incomplete Lie algebra structures

We show that the nonlinear (2+1)-dimensional Three-Wave Resonant Interaction equations, describing several important physical phenomena, can be generated starting from incomplete Lie algebras in the framework of multidimensional prolongation structures. We make use of an ansatz involving the structure equations of a principal prolongation connection induced by an admissible Bäcklund map.