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