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Mutations of the cluster variables generating the cluster algebra of type$A^{(2)}_2$ reduce to a two-dimensional discrete integrable system given by aquartic birational map. The invariant curve of the map is a singular quarticcurve, and its resolution of the singularity induces a discrete integrablesystem on a conic governed by a cubic birational map conjugate to the clustermutations of type $A^{(2)}_2$. Moreover, it is shown that the conic is also theinvariant curve of the quadratic birational map arising from the clustermutations of type $A^{(1)}_1$ and the two birational maps on the conic arecommutative. Finally, the commutative birational maps are reduced as singularlimits of additions of points on an elliptic curve arising as the spectralcurve of the discrete Toda lattice of type $A^{(1)}_1$.

Birational maps conjugate to the rank 2 cluster mutations of affine types and their geometry