Reconstruction of a homogeneous polynomial from its additive decompositions when identifiability fails

Let $X\subset {\mathbb{P}}^r$ be an integral and non-degenerate variety. For any $q\in {\mathbb{P}}^r$ let $r_X(q)$ be its $X$-rank and $\mathcal{S}(X,q)$ the set of all finite subsets of $X$ such that $\mathrm{\mid }S\mathrm{\mid }=r_X(q)$ and $q\in \langle S\rangle$,where$\langle \ \ \rangle$ denotes the linear span. We consider the case $\mathrm{\mid }\mathcal{S}(X,q)\mathrm{\mid }>1$ (i.e. when $q$ is not $X$-identifiable) and study the set $W(X{)}_q:={\cap }_{S\in \mathcal{S}}⟨S⟩$, which we call the non-uniqueness set of $q$. We study the case $\dim X=1$ and the case $X$ a Veronese embedding of ${\mathbb{P}}^n$.