  We find all the diagonal $K$-matrices for the $R$-matrix associated with the
minimal representation of the exceptional affine algebra $G^{(1)}_2$. The
corresponding transfer matrices are diagonalized with a variation of the
analytic Bethe ansatz. We find many similarities with the case of the
Izergin-Korepin $R$-matrix associated with the affine algebra $A^{(2)}_2$.
