The Fokas-Gel’fand theorem on the immersion formula of 2D-surfaces is related to the study of Lie symmetries of an integrable system. A rigorous proof of this theorem is presented which may help to better understand the immersion formula of 2D-surfaces in Lie algebras. It is shown, that even under weaker conditions, the main result of this theorem is still valid. A connection is established between three different analytic descriptions for immersion functions of 2D-surfaces, corresponding to the following three types of symmetries: gauge symmetries of the linear spectral problem, conformal transformations of the spectral parameter and generalized symmetries of the integrable system. The theoretical results are applied to the CP^{N-1} sigma model and several soliton-surfaces associated with these symmetries are constructed. It is shown that these surfaces are linked by the gauge transformations.