  Extensive elastic scattering data measured at energies around 10 to 20
MeV/nucleon for some identical systems, like 12C+12C and 16O+16O, exhibit the
nuclear rainbow pattern of broad Airy oscillations of the cross section at
medium and large angles. Due to the identity of the scattered projectile and
recoiled target, the rainbow pattern at angles around and beyond $\theta_{\rm
c.m.}\approx 90^\circ$ is strongly deteriorated by the boson exchange. The
nuclear rainbow features in the identical-particles elastic scattering
discussed so far are based on the nearside-farside (NF) decomposition of the
scattering amplitude given by an optical model calculation neglecting the
projectile-target exchange symmetry. Moreover, the NF decomposition method was
developed in the 70s by Fuller for nonidentical systems only, and the details
of how the exchange symmetry of an identical system affects the evolution of
nuclear rainbow remain unexplored. Therefore, the Fuller method is generalized
in this work for the elastic scattering of two identical (spin-zero) nuclei,
with the projectile-target exchange symmetry taken explicitly into account. The
results obtained for elastic 12C+12C and 16O+16O scattering at low energies
show the exchange symmetry results in a symmetric interchange of the nearside
and farside patterns at angles passing $\theta_{\rm c.m.}=90^\circ$, which
requires a more subtle interpretation of nuclear rainbow. We also found that a
similar NF interchange occurs in a nonidentical nucleus-nucleus system with the
core-core symmetry, where the elastic cross section at backward angles is due
mainly to the elastic transfer of cluster or nucleon between two identical
cores. This interesting effect is illustrated in the elastic 16O+12C scattering
at low energies where the elastic $\alpha$ transfer between two 12C cores has
been proven to enhance the elastic cross section at backward angles.
