  We investigate the isochronous bifurcations of the straight-line librating
orbit in the Henon-Heiles and related potentials. With increasing scaled energy
e, they form a cascade of pitchfork bifurcations that cumulate at the critical
saddle-point energy e=1. The stable and unstable orbits created at these
bifurcations appear in two sequences whose self-similar properties possess an
analytical scaling behavior. Different from the standard Feigenbaum scenario in
area preserving two-dimensional maps, here the scaling constants \alpha and
\beta corresponding to the two spatial directions are identical and equal to
the root of the scaling constant \delta that describes the geometric
progression of bifurcation energies e_n in the limit n --> infinity. The value
of \delta is given analytically in terms of the potential parameters.
