We use an algorithm based on the notion of transvectants from classical invariant theory in deter-mining the form of Stanley decomposition of the ring of invariants for the coupled Takens-Bogdanovsystems when the Stanley decompositions of the Jordan blocks of the linear part are known at eachstage. The set of systems of differential equations that are in normal form with respect to a par-ticular linear part has the structure of a module of equivariants, and is best described by giving aStanley decomposition of that module.
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