The need for an efficient method of integration of a dense normal field is inspired by several computer vision tasks, such as shape-from-shading, photo-metric stereo, deflectometry, etc. Our work is divided into two papers. In the first paper entitled Part I: A Survey, we have selected the most important properties that one may expect from any integration method. We have then reviewed most existing methods, according to the selected properties, and concluded that no method satisfies all of them. In the present paper entitled Part II: New Insights, we propose several variational methods which aim at filling this gap. We first introduce a new discretization for quadratic integration, which is designed to ensure both fast recovery and the ability to handle non-rectangular domains. Yet, with this solver, discontinuous surfaces can be handled only if the scene is first segmented into pieces without discontinuity. Inspired by edge-preserving methods from image processing , we then introduce several discontinuity-preserving functionals. Integration inspired by the Mumford-Shah segmentation method is shown to be the most effective for recovering discontinuities and kinks, and anisotropic diffusion represents a good compromise in view of the whole set of criteria.