  This note is an extension of [DZ23] there the supersymmetric vacua of
three-dimensional $\mathcal{N}=2$ gauge theories with matter are shown to be in
one-to-one correspondence with the eigenstate of $\text{XXZ}$ integrable spin
chain Hamiltonians with open boundary conditions. We consider the $A_{2}$
quiver gauge theory, which is the simplest non-trivial quiver gauge theory, and
$sl_{3}$ open $\text{XXZ}$ spin chain with diagonal boundary condition. We
demonstrate the correspondence between the vacuum equations of different gauge
groups and Bethe Ansatz equations with different boundary parameters. Not only
that, but we furthermore push forward the program to the general $A_{r}$ quiver
gauge theory.
