  In this paper we investigate complex solutions of the Bethe equations in the
two-particle sector both for arbitrary finite number of sites and for the
thermodynamic limit . We find the number of complex solutions (strings) and
compare it with the string conjecture prediction. Some simple properties of
these solutions like position in the spectrum, crossing of levels, connection
to the ground state and transformation to the real solutions are discussed.
Counting both real and complex solutions we find expected number of highest
weight Bethe states.
