  In a preprint by Montgomery \cite{Mo99}, the author attempted to prove the
existence of a shape space Figure-$8$ solution of the Newtonian three body
problem with two equal masses (it looks like a figure $8$ in the shape space,
which is different from the famous Figure-$8$ solution with three equal masses
\cite{CM00}). Unfortunately there is an error in the proof and the problem is
still open.
  Consider the $\alpha$-homogeneous Newton-type potential, $1/r^{\alpha},$
using action minimization method, we prove the existence of this solution, for
$\alpha \in (1,2)$; for $\alpha=1$ (the Newtonian potential), an extra
condition is required, which unfortunately seems hard to verify at this moment.
