  In non-Kramers Kondo systems with a quadrupolar degrees of freedom, an
ordinary magnetic Kondo effect can compete with the quadrupolar Kondo effect.
We discuss such competition keeping Pr$T_{2}$Zn$_{20}$ ($T$=Ir, Rh) and
Pr$T_{2}$Al$_{20}$ ($T$=V, Ti) in mind, where the $\Gamma_{3}$ non-Kramers
crystalline-electric-field (CEF) doublet ground state is realized in Pr$^{3+}$
ion with $(4f)^{2}$ configuration under cubic symmetry. The quadrupolar Kondo
effect can be described by the two-channel Kondo model, which leads to the
local non-Fermi-liquid (NFL) ground state, while the magnetic Kondo effect
favors the ordinary local Fermi-liquid (FL) ground state. Based on the minimal
extended two-channel Kondo model including the magnetic Kondo coupling as well,
we investigate the competition and resulting thermodynamics, and
orbital/magnetic, and single-particle excitation spectra by the Wilson's
numerical renormalization-group (NRG) method. There exists the first-order
transition between the NFL and FL ground states. In addition to these two
states, the alternative FL state accompanied by a free magnetic spin appears in
intermediate temperature range, which eventually reaches the true NFL ground
state, as a consequence of stronger competition between the magnetic and
quadrupolar Kondo effects. In this peculiar state, the magnetic susceptibility
shows a Curie-like behavior, while the orbital fluctuation exhibits the FL
behavior. Moreover, the single-particle spectra yield a more singular behavior.
Implications to the Pr 1-2-20 systems are briefly discussed.
