  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.
