  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 pseudoenergies, we show that any such system having the
same form as those for the $ADE$ TBA's, can be encoded on $A,D,E,A/Z_2$ only.
This includes, besides the known $ADE$ diagonal scattering, the set of all
$SU(2)$ related {\em magnonic} TBA's. We explore this class sistematically 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.
