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Group classification of the general second-order evolution equation: semi-simple invariance groups

In this paper, we consider the problem of group classification of the generic second-order evolution equation in one spatial variable. We construct all inequivalent evolution equations whose invariance groups are either semi-simple or semi-direct products of semi-simple and solvable Lie groups. The obtained lists of invariant equations contain both already known equations and the broad classes of new evolution equations possessing nontrivial Lie symmetry.