Solution of tetrahedron equation and cluster algebras

We notice a remarkable connection between Bazhanov-Sergeev solution of Zamolodchikov tetrahedron equation and certain well-known cluster algebra expression. The transformation is then identified with sequence four mutations. As an application the new formalism we show how to construct integrable system spectral curve arbitrary symmetric Newton polygon. Finally, embed this into double Bruhat cell Poisson-Lie group, triangular decomposition can be used extend our approach general non-symmetric polygons, prove Lemma, which classifies conjugacy classes in affine Weyl groups $A$-type by polygons.