  For a group $G$ acting on an affine variety $X$, the separating variety is
the closed subvariety of $X\times X$ encoding which points of $X$ are separated
by invariants. We concentrate on the indecomposable rational linear
representations $V_n$ 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.
