Abstract
The symmetries, especially those related to the R-transformation, of the reflection equation (RE) for two-component systems are analysed. 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 a Baxter type corresponds to the same form of a 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 is obtained with a generalization to the Sklyanin formalism.