  We propose a non-standard separation of variables for the classical
integrable XXX and XXZ spin chains with degenerate twist matrix. We show that
for the case of such twist matrices one can interchange the role of classical
separating functions $A(u)$ and $B(u)$ and construct a new full set of
separated variables, satisfying simpler equation of separation and simpler Abel
equations in comparison with the standard separated variables of Sklyanin. We
show that for certain cases of the twist matrices the constructed separated
variables can be directly identified with action-angle coordinates.
