  Among other results we show that near the equilibrium point, the Hamiltonian
of the sine-Gordon (SG) equation on the circle can be viewed as an element in
the Poisson algebra of the modified Korteweg-de Vries (mKdV) equation and hence
by well established properties of the latter equation admits Birkhoff
coordinates. On the other hand we prove that there exists a large set of smooth
initial data, away from the equilibrium point and lying on the ramification
locus of a double cover, for which the initial value problem of the SG equation
has no classical solution.
