Pedro has studied the origin, stability, and bifurcation structure of different types of bright localized structures described by the Lugiato-Lefever equation. This mean field model describes the nonlinear dynamics of light circulating in fiber cavities and microresonators. In this work, published in Phys. Rev. E , we show how the organization (bifurcation structure) of these localized light patterns critically changes as one increases the cavity detuning, one of the important experimental control parameters in the system.