An extended representation of three spin component Bose Einstein Condensate solitons

We consider a three spin component Bose Einstein Condensate as described by as many coupled nonlinear Schroedinger equations. For a very special ratio of the coupling constants, exact N soliton solutions to this set of equations are known. Here we find a simple representation including the N = 1 solution based on the symmetry of the equations. This symmetry is described by a linear operator, the nonlinearity of NLS notwithstanding. Our useful representation opens the door to the nonintegrable case of general coupling constants. A new class of solutions is found.