  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 $A^{(1)}_1$.
