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

Mutations of the cluster variables generating the cluster algebra of type $A_2^{(2)}$ reduce to a two-dimensional discrete integrable system given by a quartic birational map. The invariant curve of the map is a singular quartic curve, and its resolution of the singularity induces a discrete integrable system on a conic governed by a cubic birational map conjugate to the cluster mutations of type $A_2^{(2)}$. Moreover, it is shown that the conic is also the invariant curve of the quadratic birational map arising from the cluster mutations of type $A_1^{(1)}$ and the two birational maps on the conic are commutative. Finally, the commutative birational maps are reduced as singular limits of additions of points on an elliptic curve arising as the spectral curve of the discrete Toda lattice of type $A_1^{(1)}$.