Classification of KdV vessels with constant parameters and two dimensional outer space

Abstract: 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 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).