The symmetries, especially those related to the $R$-transformation, of the
reflection equation(RE) for two-component systems are analyzed. The
classification of solutions to the RE for eight-, six- and seven-vertex type
$R$-matrices is given. All solutions can be obtained from those corresponding
to the standard $R$-matrices by $K$-transformation. For the free-Fermion
models, the boundary matrices have property $tr K_+(0)=0$, and the free-Fermion
type $R$-matrix with the same symmetry as that of Baxter type corresponds to
the same form of $K_-$-matrix for the Baxter type. We present the Hamiltonians
for the open spin systems connected with our solutions. In particular, the
boundary Hamiltonian of seven-vertex models was obtained with a generalization
to the Sklyanin's formalism.