  In this work, we systematically study the $D\bar{D}^*$/$B\bar{B}^*$ and
$DD^*$/$\bar{B}\bar{B}^*$ systems with the Bethe-Salpeter equation in the
ladder and instantaneous approximations for the kernel. By solving the
Bethe-Salpeter equation numerically with the kernel containing the direct and
crossed one particle-exchange diagrams and introducing three different form
factors (monopole, dipole, and exponential form factors) at the vertices, we
find only the isoscalar $D\bar{D}^*$/$B\bar{B}^*$ and $DD^*$/$\bar{B}\bar{B}^*$
systems can exist as bound states. This indicate that the $X(3872)$ and
$T^+_{cc}$ could be accommodated as $I^G(J^{PC})=0^+(1^{++})$ $D\bar{D}^*$ and
$(I)J^P =(0)1^+$ $DD^*$ molecular states while the molecular state explanations
for $Z_b(10610)$ and $Z_c(3900)$ are excluded.
