Abstract We develop an inverse scattering transformation in the angular momentum to deal with rotationally invariant problems in higher dimensions. We consider the two-dimensional non-linear σ-model in the 1 N expansion. We succeed in expressing the action (a renormalized determinant) in terms of spectral data and we show that no real saddle points exist. This relates to an instability under dilatations connected with the asymptotic freedom of the model. This result together with Zamolodchikov's S-matrix lead us to conjecture that the 1 N series may be convergent for this model.