  In this article we classify vessels producing solutions of some completely
integrable PDEs, presenting a \textit{unified} approach for them. The
classification includes such important examples as Korteweg-de Vries (KdV) and
evolutionary Non Linear Schr\" odingier (ENLS) equations. In fact, employing
basic matrix algebra techniques it is shown that there are exactly two
canonical forms of such vessels, so that each canonical form generalize either
KdV or ENLS equations. Particularly, Dirac canonical systems, whose evolution
was recently inserted into the vessel theory, are shown to be equivalent to the
ENLS equation in the sense of vessels. This work is important as a first step
to classification of completely integrable PDEs, which are solvable by the
theory of vessels. We note that a recent paper of the author, published in
Journal of Mathematical Physics, showed that initial value problem with
analytic initial potential for the KdV equation has at least a "narrowing" in
time solution. The presented classification, inherits this idea and a similar
theorem can be easily proved for the presented PDEs. Finally, the the resuts of
the work serve as a basis for the investigation of the following problems: 1.
hierarchy of the generalized KdV, ENLS equations (by generalizing the vessel
equations), 2. new completely integrable PDEs (by changing the dimension of the
outer space), 3. addressing the question of integrability of a given arbitrary
PDE (the future classification will create a list of solvable by vessels
equations, which may eventually include many existing classes of PDEs).
