  This work concerns to the studies of boundary integrability of the vertex
models from representations of the Temperley-Lieb algebra associated with the
quantum group ${\cal U}_{q}[X_{n}]$ for the affine Lie algebras
  $X_{n}$ = $A_{1}^{(1)}$, $B_{n}^{(1)}$, $C_{n}^{(1)}$ and $D_{n}^{(1)}$.
  A systematic computation method is used to constructed solutions of the
boundary Yang-Baxter equations. We find a $2n^{2}+1$ free parameter solution
for $A_{1}^{(1)} $ spin-$(n-1/2)$ and $ C_{n}^{(1)}$ vertex models. It turns
that for $A_{1}^{(1)} $ spin-$n$, $ B_{n}^{(1)}$ and $D_{n}^{(1)}$ vertex
models, the solution has $2n^{2}+2n+1$ free parameters.
