  The hidden E_{7} (E_{6}) structure has been conjectured for the minimal model
${\cal M}_{4,5} ({\cal M}_{6,7}$) perturbed by $\Phi_{1,2}$ in the context of
conformal field theory(CFT). Motivated by this, we examine the dilute A_{4, 6}
models, which are expected to be corresponding lattice models. Thermodynamics
of the equivalent one dimensional quantum systems is analyzed via the quantum
transfer matrix approach. Appropriate auxiliary functions, related to kinks in
the theory, play a role in constructing functional relations among transfer
matrices. We successfully recover the universal Y- systems and thereby
Thermodynamic Bethe Ansatz equations for E_{6,7} from the dilute A_{6,4} model,
respectively.
