  We show that solitonic solutions of the classical string action on the AdS_5
x S^5 background that carry charges (spins) of the Cartan subalgebra of the
global symmetry group can be classified in terms of periodic solutions of the
Neumann integrable system. We derive equations which determine the energy of
these solitons as a function of spins. In the limit of large spins J, the first
subleading 1/J coefficient in the expansion of the string energy is expected to
be non-renormalised to all orders in the inverse string tension expansion and
thus can be directly compared to the 1-loop anomalous dimensions of the
corresponding composite operators in N=4 super YM theory. We obtain a closed
system of equations that determines this subleading coefficient and, therefore,
the 1-loop anomalous dimensions of the dual SYM operators. We expect that an
equivalent system of equations should follow from the thermodynamic limit of
the algebraic Bethe ansatz for the SO(6) spin chain derived from SYM theory. We
also identify a particular string solution whose classical energy exactly
reproduces the one-loop anomalous dimension of a certain set of SYM operators
with two independent R charges J_1, J_2.
