The Margulis invariant α is a function on H
1(Γ,ℝ2,1), where Γ is a group of Lorentzian transformations acting on ℝ2,1, that contains no elliptic elements. The spectrum of Γ is the image of all γ∈Γ∖ (Id) under the map α. If the underlying linear group of Γ is fixed, Drumm and Goldman proved that the spectrum defines the translational part completely. In this note, we strengthen this result by showing that isospectrality holds for any free product of cyclic groups of given rank, up to conjugation in the group of affine transformations of ℝ2,1, as long as it is non-radiant, and that its linear part is discrete and non-elementary. In particular, isospectrality holds when the linear part is a Schottky group.