Abstract
In this article we suggest an improved method for classifying general two-component integrable evolution equations, homogeneous in a given weighting scheme. This method relies on linear changes of variables and on an appropriate splitting of the solution space. To illustrate the method, we implement the classification of coupled KdV-type equations. We show that there are five nontriangular systems possessing higher order generalized symmetries. One of these systems is previously unknown. This seems to be the first classification of coupled integrable equations homogeneous in a given weighting scheme, without any restrictions on the form of the main matrix.
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