  We completely describe the Brill-Noether theory for curves in the primitive
linear system on generic abelian surfaces, in the following sense: given
integers $d$ and $r$, consider the variety $V^r_d(|H|)$ parametrizing curves
$C$ in the primitive linear system $|H|$ together with a torsion-free sheaf on
$C$ of degree $d$ and $r+1$ global sections. We give a necessary and sufficient
condition for this variety to be non-empty, and show that it is either a
disjoint union of Grassmannians, or irreducible. Moreover, we show that, when
non-empty, it is of expected dimension.
  This completes prior results by Knutsen, Lelli-Chiesa and Mongardi.
