Let X ⊂ ℙ r be an integral and non-degenerate complex variety. For any q ∈ ℙ r let r X ðqÞ be its X-rank and SðX ; qÞ the set of all finite subsets of X such that jSj ¼ r X ðqÞ and q ∈ hSi, where h i denotes the linear span. We consider the case jSðX ; qÞj > 1 (i.e. when q is not X -identifiable) and study the set W ðX Þ q :¼ ∩ S∈SðX ;qÞ hSi, which we call the non-uniqueness set of q. We study the case dim X ¼ 1 and the case X a Veronese embedding of ℙ n . We conclude the paper with a few remarks concerning this problem over the reals.