  We revisit in this note the H\'enon's isochrone problem. By using the
standard Abel inversion technique for one-dimensional motion, we recover in a
simple way the H\'enon's parabolae and get all isochrone central potentials
under mild smoothness assumptions on the potential function. Our approach also
allows us to conclude that isochronous radial periods with explicit energy
dependence are necessarily Keplerian, i.e., $T^{2}\propto|E|^{-3}$, and that
their corresponding orbits can be easily integrated by mapping them into the
usual Kepler problem. It can also be employed to study some other inverse
central-force problems and, in particular, it provides a proof of Bertrand's
theorem.
