  In this paper, we first describe how we can arrange any bodies on
Figure-Eight without collision in a dense subset of $[0,T]$ after showing that
the self-intersections of Figure-Eight will not happen in this subset. Then it
is reasonable for us to consider the existence of generalized solutions and
non-collision solutions with Mixed-symmetries or with Double-Eight constraints,
arising from Figure-Eight, for N-body problem. All of the orbits we found
numerically in Section \ref{se7} have not been obtained by other authors as far
as we know. To prove the existence of these new periodic solutions, the
variational approach and critical point theory are applied to the classical
N-body equations. And along the line used in this paper, one can construct
other symmetric constraints on N-body problems and prove the existence of
periodic solutions for them.
