  In this paper we study cluster algebras $\myAA$ of type $A_2^{(1)}$. We solve
the recurrence relations among the cluster variables (which form a T--system of
type $A_2^{(1)}$). We solve the recurrence relations among the coefficients of
$\myAA$ (which form a Y--system of type $A_2^{(1)}$). In $\myAA$ there is a
natural notion of positivity. We find linear bases $\BB$ of $\myAA$ such that
positive linear combinations of elements of $\BB$ coincide with the cone of
positive elements. We call these bases \emph{atomic bases} of $\myAA$. These
are the analogue of the "canonical bases" found by Sherman and Zelevinsky in
type $A_{1}^{(1)}$. Every atomic basis consists of cluster monomials together
with extra elements. We provide explicit expressions for the elements of such
bases in every cluster. We prove that the elements of $\BB$ are parameterized
by $\ZZ^3$ via their $\mathbf{g}$--vectors in every cluster. We prove that the
denominator vector map in every acyclic seed of $\myAA$ restricts to a
bijection between $\BB$ and $\ZZ^3$. In particular this gives an explicit
algorithm to determine the "virtual" canonical decomposition of every element
of the root lattice of type $A_2^{(1)}$. We find explicit recurrence relations
to express every element of $\myAA$ as linear combinations of elements of
$\BB$.
