This paper presents a derivation of the possible residual symmetries of rational K-matrices which are invertible in the ''classical limit'' (the spectral parameter goes to infinity). This derivation uses only the boundary Yang-Baxter equation and the asymptotic expansions of the R-matrices. The result proves the previous assumption of the literature: if the original and the residual symmetry algebras are $\mathfrak{ g}$ and $\mathfrak{ h}$ then there exists a Lie-algebra involution of $\mathfrak{ g}$ for which the invariant sub-algebra is $\mathfrak{ h}$. In addition, we study some K-matrices which are not invertible in the ''classical limit''. It is shown that their symmetry algebra is not reductive but a semi-direct sum of reductive and solvable Lie-algebras.
Categories: math-ph hep-th math.MP nlin.SI