  Recent progress in two different fronts is reported. First, the concept of
bisection of a harmonic oscillator (HO) or hydrogen atom (HA), used in the past
in establishing the connection between U(3) and O(4), is generalized into
multisection (trisection, tetrasection, etc). It is then shown that all
symmetries of the N-dim anisotropic HO with rational ratios of frequencies
(RHO), some of which are underlying the structure of superdeformed and
hyperdeformed nuclei, can be obtained from the U(N) symmetry of the
corresponding isotropic HO with an appropriate combination of multisections.
Furthermore, it is seen that bisections of the N-dim HA, which possesses an
O(N+1) symmetry, lead to the U(N) symmetry, so that further multisections of
the HA lead to the symmetries of the N-dim RHO. Second, it is shown that there
is evidence that the recently observed in superdeformed nuclear bands $\Delta
I=4$ bifurcation is also occuring in rotational bands of diatomic molecules. In
addition there is evidence that a $\Delta I=8$ bifurcation, of the same order
of magnitude as the $\Delta I=4$ one, is observed in superdeformed nuclear
bands and rotational bands of diatomic molecules.
