  The Margulis invariant is a function defined on a group of Lorentzian
transformations $G$ acting on Minkowski space $\R^{2,1}$, that contains no
elliptic elements. The spectrum of $G$ is the sequence of values of the
Margulis invariant for all its elements. If the underlying linear group of $G$
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 $R^{2,1}$, 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.
