  We define regularised Poisson brackets for the monodromy matrix of classical
string theory on R x S^3. The ambiguities associated with Non-Ultra Locality
are resolved using the symmetrisation prescription of Maillet. The resulting
brackets lead to an infinite tower of Poisson-commuting conserved charges as
expected in an integrable system. The brackets are also used to obtain the
correct symplectic structure on the moduli space of finite-gap solutions and to
define the corresponding action-angle variables. The canonically-normalised
action variables are the filling fractions associated with each cut in the
finite-gap construction. Our results are relevant for the leading-order
semiclassical quantisation of string theory on AdS_5 x S^5 and lead to
integer-valued filling fractions in this context.
