  In the strong coupling limit, an effective resonant level model is derived
for the spin-1 underscreened Kondo impurity model. A local quantum critical
behavior is induced by the formation of a bound state with partially screened
magnetic moment, displaying a residual $Z_2$ symmetry, leading to a $\delta
$-resonance at the Fermi level in the impurity spectral function. As a
consequence, a logarithmic singularity appears in the real part of the impurity
dynamic spin susceptibility as a function of $\max (\omega, T)$, with $\omega $
as frequency, $T$ temperature. A small magnetic field breaks this $Z_2$
symmetry and suppresses the singularity. We also discuss the possible
manifestation of such a quantum critical behavior in the spin-1/2 two Kondo
impurity model and the single quantum dot system with even electron occupation.
