Abstract
The authors give new linearisation equations for the O(3) new-linear σ-model which are similar to those obtained in the stationary axially symmetric Einstein field equations. By using the linearisation equations, they confirm the existence of the infinite-dimensional symmetries, which are dependent on a spectrum parameter, in the O(3) non-linear σ-model. The relationships between the hidden transformations and the infinitesimal Riemann-Hilbert transformation are discussed.