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We prove a useful identity valid for all ADE minimal S-matrices, that clarifies the transformation of the relative thermodynamic Bethe Ansatz (TBA) from its standard form into the universal one proposed by Al. B. Zamolodchikov. By considering the graph encoding of the system of functional equations for the exponentials of the pseudo-energies, we show that any such system having the same form as those for the ADE TBA’s, can be encoded 011 A, D, E, A/Z₂ only. This includes, besides the known ADE diagonal scattering, the set of all SU(2) related magnonic TBA’s. We explore this class systematically and find some interesting new massive and massless RG flows. The generalization to classes related to higher rank algebras is briefly presented and an intriguing relation with level-rank duality is signalled.