  The ``truncation procedure'' initiated by Weiss et al. is best understood as
a Darboux transformation. If it leads to the Lax pair of the PDE under study,
the B\"acklund transformation follows by an elimination, thus proving the
integrability. We present the state of the art of this powerful technique. The
easy situations were all handled by the WTC one-family truncation and its
homographically invariant version. An updated version of this method has been
recently developed, which is now able to handle the Kaup-Kupershmidt and
Tzitz\'eica equations. It incorporates a new feature, namely the distinction
between two entire functions usually mingled, which are shown to be linked by
formulae established by Gambier for his classification.
