  Studying perturbatively, for large m, the torus partition function of both
(A,A) and (A,D) series of minimal models in the Cappelli, Itzykson, Zuber
classification, deformed by the least relevant operator $\phi_{(1,3)}$, we
disentangle the structure of $\phi_{1,3}$ flows. The results are conjectured on
reasonable ground to be valid for all m. They show that (A,A) models always
flow to (A,A) and (A,D) ones to (A,D). No hopping between the two series is
possible. Also, we give arguments that there exist 3 isolated flows
(E,A)-->(A,E) that, together with the two series, should exhaust all the
possible $\phi_{1,3}$ flows. Conservation (and symmetry breaking) of non-local
currents along the flows is discussed and put in relation to the A,D,E
classification.
