  In this paper we study the behavior of the Ricci flow at infinity for the
full flag manifold $SU(3)/T$ using techniques of the qualitative theory of
differential equations, in special the Poincar\'e Compactification and Lyapunov
exponents. We prove that there are four invariant lines for the Ricci flow
equation, each one associated with a singularity corresponding to a Einstein
metric. In such manifold, the bi-invariant normal metric is Einstein. Moreover,
around each invariant line there is a cylinder of initial conditions such that
the limit metric under the Ricci flow is the corresponding Einstein metric; in
particular we obtain the convergence of left-invariant metrics to a
bi-invariant metric under the Ricci flow.
