  The flavor-mixing induced by the mismatched vector-isoscalar and
vector-isovector interactions at finite baryon chemical potential $\mu$ and
isospin chemical potential $\mu_I$ is demonstrated in the Nambu-Jona-Lasinio
(NJL) type model of QCD. The influence of this non-anomaly flavor-mixing on the
possible separate chiral transitions at nonzero $\mu_I$ is studied under the
assumption of the effective restoration of the $U(1)_A$ symmetry. We find that
for the weak isospin asymmetry, the two separate phase boundaries found
previously can be converted into one only if the vector-isovector coupling
$g_v^v$ is significantly stronger than the vector-isoscalar one $g_v^s$ without
the axial anomaly. When the weak Kabayashi-Maskawa-'t Hooft (KMT) interaction
is included, we find that the separation of the chiral transition with two
critical endpoints for the relatively strong isospin asymmetry can still be
removed owning to the vector interactions. In this case, it is not the vector
coupling difference but the strength of $g_v^v$ which is crucial for the only
phase boundary. We also point out that, in the NJL-type model with mismatched
vector interactions, the recently proposed equivalence for chiral transitions
at finite $\mu$ and $\mu_I$ does not hold even at the mean field approximation.
