Zamolodchikov periodicity is periodicity of certein recursions associated with box products $X \square Y$ of two finite type Dynkin diagrams. We suggest an affine analog of Zamolodchikov periodicity, which we call Zamolodchikov integrability. We conjecture that it holds for products $X \square Y$, where $X$ is a finite type Dynkin diagram and $Y$ is an extended Dynkin diagram. We prove this conjecture for the case of $A_m \square A_{ 2n-1}^{ (1)}$. The proof employs cluster structures in certain classical rings of invariants, previously studied by S. Fomin and the author.
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