Riccati pseudopotential of AKNS two-family NLPDEs by Painlevé analysis

Finding the Bäcklund transformation from the singularity structure only is impossible with the method of Weiss if the PDE has two families of singularities with opposite principal parts, such as the modified KdV and sine-Gordon equations. For such PDEs, we first consider the truncation with one manifold, derive the Darboux transformation (DT) and show that it involves the two entire functions associated to each family. Their ratio is then assumed to satisfy the most general Riccati system. This hypothesis combined with the DT generates a very small number of determining equations, admitting a unique solution, equivalent to the matricial Lax pair by the usual linearization of the Riccati system.