Abstract: We study two problems about Bonnet surfaces. First, we prove that the Bonnet surface whose Gaussian curvature is not identically zero must exist by studying the ordinary differential equation which the mean curvature of the Bonnet surface satisfies. Secondly, we prove that if there exists a conformal map which preserves the principal curvatures and the orientation between two Bonnet surfaces, then there are two cases as follows:1) If the zero points of the Gaussian curvature of the two surfaces are isolated, then the conformal map must be an isometry. 2) If the Gaussian curvature of the two surfaces is identically zero, then the conformal map is a similarity transformation.

Key words: Bonnet surface, W-surface, conformal map, similarity transformation