In this work we study renormalization-group flows by deforming a class of conformal sigma-models. At leading order in $\alpha’$, renormalization-group equations represent a Ricci flow. In the three-sphere background, the latter is described by the Halphen system, which is exactly solvable in terms of modular forms. The round sphere is found to be the unique perturbative infra-red fixed point at one loop order.