  The quantum integrable system associated with the $G_2$ exceptional Lie
algebra is studied. By using the fusion technique, we obtain the closed
recursive relations among the fused transfer matrices. Based on the polynomial
analysis, we obtain the exact energy spectrum and the associated Bethe ansatz
equations of the system. Both the periodic and the non-diagonal open boundary
conditions are studied in a unified way. For the later case, we also obtain the
crossing symmetries of the transfer matrix and the fused ones.
