  While extending a famous problem asked and solved by Bertrand in 1873,
Darboux found in 1877 a family of abstract surfaces of revolution, each endowed
with a force function, with the striking property that all the orbits are
periodic on open sets of the phase space. We give a description of this family
which explains why they have this property: they are the Darboux inverses of
the Kepler problem on constant curvature surfaces. What we call the Darboux
inverse was briefly introduced by Darboux in 1889 as an alternative approach to
the conformal maps that Goursat had just described.
