On Computer-assisted Classification of Coupled Integrable Equations

We show how the triangularization method of Moreno Maza can be successfully applied to the problem of classification of homogeneous coupled integrable equations. The classifications rely on the recent algorithm developed by Foursov that requires solving 17 systems of polynomial equations. We show that these systems can be completely resolved in the case of coupled Korteweg–de Vries, Sawada–Kotera and Kaup–Kupershmidt-type equations.