  We present studies of a simple Anderson model Hamiltonian for Ce$^{3+}$ ions
in cubic symmetry with three configurations ($f^0$,$f^1$,$f^2$). Our
Hamiltonian includes: (i) One-channel Anderson model; (ii) Two-channel Anderson
model. Using the third order scaling analysis, we study stability of the
non-Fermi liquid fixed point of the two-channel Kondo model for Ce$^{3+}$ ions
in cubic symmetry against the one-channel Kondo interaction. Using the
non-crossing approximation (NCA), we also report detailed studies of our
simplified model of one-channel \& two-channel Anderson model which exhibits
competition between the Fermi-liquid fixed point of the one-channel Kondo model
and the non-Fermi fixed point of the two-channel Kondo model. We provide the
phase diagram in the model parameter space and study the thermodynamics and the
transport properties of our simplified model Hamiltonian. Thermodynamics and
transport coefficients show the distinct behaviors for different numbers of
channels. We confirm in detail that the NCA is a valid numerical method for the
overcompensated multichannel $S_I=1/2$ Anderson models. Our model study might
be relevant to the non-Fermi liquid alloy Ce$_{1-x}$La$_x$Cu$_{2.2}$Si$_2$.
