  In this paper we find a class of new degenerate central configurations and
bifurcations in the Newtonian $n$-body problem. In particular we analyze the
Rosette central configurations, namely a coplanar configuration where $n$
particles of mass $m_1$ lie at the vertices of a regular $n$-gon, $n$ particles
of mass $m_2$ lie at the vertices of another $n$-gon concentric with the first,
but rotated of an angle $\pi/n$, and an additional particle of mass $m_0$ lies
at the center of mass of the system. This system admits two mass parameters
$\mu=m_0/m_1$ and $\ep=m_2/m_1$. We show that, as $\mu$ varies, if $n> 3$,
there is a degenerate central configuration and a bifurcation for every
$\ep>0$, while if $n=3$ there is a bifurcations only for some values of
$\epsilon$.
