  In a remarkable paper of 2003 by Fujiwara et al. \cite{Fujiwara2003}, a
figure-eight three-body choreography on the algebraic lemniscate by Bernoulli
was discovered. Such a choreography was found to be driven by the action of a
pairwise potential $ V(r_{ij}) $, depending only on the mutual relative
distances $r_{ij}, i,j=1,2,3$. In the present Letter we show that two different
choreographies of five bodies on the same algebraic lemniscate exist and
correspond to solutions of ten coupled Newton equations of motion with a
pairwise interaction potential. For each choreography the explicit form of the
potential is found and ten constants of motion are presented, thus, it is
superintegrable.
