  We study the dynamics of finite-gap solutions in classical string theory on R
x S^3. Each solution is characterised by a spectral curve, \Sigma, of genus g
and a divisor, \gamma, of degree g on the curve. We present a complete
reconstruction of the general solution and identify the corresponding
moduli-space, M^(2g)_R, as a real symplectic manifold of dimension 2g. The
dynamics of the general solution is shown to be equivalent to a specific
Hamiltonian integrable system with phase-space M^(2g)_R. The resulting
description resembles the free motion of a rigid string on the Jacobian torus
J(\Sigma). Interestingly, the canonically-normalised action variables of the
integrable system are identified with certain filling fractions which play an
important role in the context of the AdS/CFT correspondence.
