The separating variety for the basic representations of the additive group

For a group G acting on an affine variety X, the separating variety is the closed subvariety of X×X encoding which points of X are separated by invariants. We concentrate on the indecomposable rational linear representations Vn of dimension n+1 of the additive group of a field of characteristic zero, and decompose the separating variety into the union of irreducible components. We show that if n is odd, divisible by four, or equal to two, the closure of the graph of the action, which has dimension n+2, is the only component of the separating variety. In the remaining cases, there is a second irreducible component of dimension n+1. We conclude that in these cases, there are no polynomial separating algebras.