Abstract
In this paper we describe a new method for constructing integrable systems of differential equations. We are looking for systems in two variables in such forms that the reduction v = u leads us to a single equation in u. We give a complete classification of such systems that reduce to Korteweg-de Vries-type equations. Furthermore, we present an extensive (and complete for the systems of the Sawada-Kotera and Kaup-Kupershmidt types) classification of fifth-order equations in the same weighting. We show that the scalar integrable equations give rise to large classes of integrable systems. Moreover, we present a previously unknown example of a system that can be written in biHamiltonian form in infinitely many different ways, thereby solving the problem of the number of biHamiltonian forms that can have a differential equation. Finally, we present examples of nondegenerate systems possessing degenerate symmetries, which is impossible in the scalar case.
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