  To study the impurity potential effect to the Kondo problem in a Luttinger
liquid, we propose an integrable model of two interacting half-chains coupled
with a single magnetic impurity ferromagnetically. It is shown that the scalar
potential effectively reconciles the spin dynamics at low temperatures.
Generally, there is a competition between the Kondo coupling $J$ and the
impurity potential $V$. When the ferromagnetic Kondo coupling dominates over
the impurity potential ($V<|SJ|$), the Furusaki-Nagaosa many-body singlet can
be perfectly realized. However, when the impurity potential dominates over the
Kondo coupling ($V\geq |SJ|$), the fixed point predicted by Furusaki and
Nagaosa is unstable and the system must flow to a weak coupling fixed point. It
is also found that the effective moment of the impurity measured from the
susceptibility is considerably enlarged by the impurity potential. In addition,
some quantum phase transitions driven by the impurity potential are found and
the anomaly residual entropy is discussed.
