Abstract
The foundations of the symmetry approach to the classification problem of integrable non-linear evolution systems are briefly described. Within the framework of the symmetry approach the ten-parametric family of the third order non-linear evolution coupled KdV-like systems is investigated. The necessary integrability conditions lead to an over-determined non-linear algebraic system. To solve that system an effective method based on its structure has been used. This allows us to obtain the complete list of integrable systems of a given type. All computation has been completed on the basis of computer algebra systems FORMAC and REDUCE.
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