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This article aims to carry out novel research on obtaining exact solutions to a generalized time-fractional KdV-Drinfel’d-Sokolov-Wilson system. The complete discrimination system for polynomial method is applied to handle such a problem. Description of the method is given and all the possible exact traveling wave solutions including hyper-elliptic functions solutions are obtained, which are very difficult to get by other methods. In particular, to show the existence of the solutions, concrete examples are constructed for the concrete parameters of the corresponding solutions, from which we can see that each kind of the solutions in the classification can be realized except one. Keywords: Exact solutions, Complete discrimination system for polynomial method, Time-fractional KdV-Drinfel’d-Sokolov-Wilson system, Traveling wave solutions