The geometry and properties of the fitness landscapes of Computational Protein Design (CPD) are not well understood, due to the difficulty for sampling methods to access the NP-hard optima and explore their neighborhoods. In this paper, we use a state-of-the-art AI complete algorithm to enumerate all solutions within a 2 kcal/mol energy interval of the optimum of two CPD problems. We compute the number of local minima, the size of the attraction basins, and the local optima network. We provide various features in order to characterize the fitness landscapes, in particular the multimodality, and the ruggedness of the fitness landscape. Results show some key differences in the fitness landscapes and help to understand the successes and failures of metaheuristics on CPD problems. Our analysis gives some previously inaccessible and valuable information on the problem structure related to the optima of the CPD instances (multi-funnel structure), and could lead to the development of more efficient metaheuristic methods.