This paper deals with the non-uniqueness of the solutions of an analysis—Lasso regularization. Most previous works in this area are concerned with the case, where the solution set is a singleton, or to derive guarantees to enforce uniqueness. Our main contribution consists in providing a geometrical interpretation of a solution with a maximal analysis support: such a solution abides in the relative interior of the solution set. Our result allows us to provide a way to exhibit a maximal solution using a primal-dual interior point algorithm.