  We present a scattering model for nuclei with similar masses. In this
three-body model, the projectile has a core+valence structure, whereas the
target is identical to the core nucleus. The three-body wave functions must be
symmetrized for the exchange of the cores. This property gives rise to
non-local potentials, which are computed without approximation. The present
model is an extension of the Continuum Discretized Coupled Channel (CDCC)
formalism, with an additional treatment of core exchange. We solve the
coupled-channel system, including non-local terms, by the $R$-matrix method
using Lagrange functions. This model is applied to the $^{13}{\rm C}+^{12}$C,
$^{13}{\rm N}+^{12}$C and $^{16}{\rm O}+^{12}$C systems. Experimental
scattering cross sections are fairly well reproduced without any parameter
fitting. The backward-angle enhancement of the elastic cross sections is due to
the non-local potential. We discuss in more detail the various non-local
contributions and present effective local potentials.
