  In this work, we investigate the bifurcations of relative equilibria in the
gravitational potential of asteroids. A theorem concerning a conserved
quantity, which is about the eigenvalues and number of relative equilibria, is
presented and proved. The conserved quantity can restrict the number of
non-degenerate equilibria in the gravitational potential of an asteroid. It is
concluded that the number of non-degenerate equilibria in the gravitational
field of an asteroid varies in pairs and is an odd number. In addition, the
conserved quantity can also restrict the kinds of bifurcations of relative
equilibria in the gravitational potential of an asteroid when the parameter
varies. Furthermore, studies have shown that there exist transcritical
bifurcations, quasi-transcritical bifurcations, saddle-node bifurcations,
saddle-saddle bifurcations, binary saddle-node bifurcations, supercritical
pitchfork bifurcations, and subcritical pitchfork bifurcations for the relative
equilibria in the gravitational potential of asteroids. It is found that for
the asteroid 216 Kleopatra, when the rotation period varies as a parameter, the
number of relative equilibria changes from 7 to 5 to 3 to 1, and the
bifurcations for the relative equilibria are saddle-node bifurcations and
saddle-saddle bifurcations.
