  We introduce various exchange interactions in a model Hamiltonian for
Ce$^{3+}$ ions in cubic symmetry with three configurations ($f^0$,$f^1$,$f^2$).
With the impurity pseudo spin $S_I=1/2$, our Hamiltonian includes: (i)
One-channel $S_c=1/2$ Anderson model; (ii) Two-channel $S_c=1/2$ Anderson
model; (iii) An unforseen one-channel $S_c=3/2$ Anderson model with a
non-trivial fixed point; (iv) Mixing exchange interaction between the
$\Gamma_{6,7}$ and the $\Gamma_8$ conduction electron partial wave states; (v)
Multiple conduction electron partial wave states. Using the third-order scaling
(perturbative renormalization group) analysis, we study stability of various
fixed points relevant to various exchange interactions for Ce$^{3+}$ ions in
cubic symmetry.
