  Given two positive real numbers $M$ and $m$ and an integer $n>2$, it is well
known that we can find a family of solutions of the $(n+1)$-body problem where
the body with mass $M$ stays put at the origin and the other $n$ bodies, all
with the same mass $m$, move on the $x$-$y$ plane following ellipses with
eccentri\-city $e$. It is expected that this geometrical family that depends on
$e$, has some bifurcations that produce solutions where the body in the center
moves on the $z$-axis instead of staying put in the origin. By doing an
analytic continuation of a periodic numerical solution of the $4$-body problem
--the one displayed on the video http://youtu.be/2Wpv6vpOxXk --we surprisingly
discovered that the origin of this periodic solution is not part of the
geometrical family of elliptical solutions parametrized by the eccentricity
$e$. It comes from a not so geometrical but easier to describe family. Having
noticed this new family, the authors find an exact formula for the bifurcation
point in this new family and use it to show the existence of a non-planar
periodic solution for any pair of masses $M$, $m$, and any integer $n$. As a
particular example, we find a solution where three bodies with mass $3$ move
around a body with mass $7$ that moves up and down.
