  We reduce the dynamics of an ensemble of mean-coupled Stuart-Landau
oscillators close to the synchronized solution. In particular, we map the
system onto the center manifold of the Benjamin-Feir instability, the
bifurcation destabilizing the synchronized oscillation. Using symmetry
arguments, we describe the structure of the dynamics on this center manifold up
to cubic order, and derive expressions for its parameters. This allows us to
investigate phenomena described by the Stuart-Landau ensemble, such as
clustering and cluster singularities, in the lower-dimensional center manifold,
providing further insights into the symmetry-broken dynamics of coupled
oscillators. We show that cluster singularities in the Stuart-Landau ensemble
correspond to vanishing quadratic terms in the center manifold dynamics. In
addition, they act as organizing centers for the saddle-node bifurcations
creating unbalanced cluster states as well for the transverse bifurcations
altering the cluster stability. Furthermore, we show that bistability of
different solutions with the same cluster-size distribution can only occur when
either cluster contains at least $1/3$ of the oscillators, independent of the
system parameters.
