  We derive exactly the number of complex solutions with two down-spins in the
massive regime of the periodic spin-1/2 XXZ spin chain of $N$ sites. Here we
remark that every solution of the Bethe ansatz equations is characterized by a
set of quantum numbers. We derive them analytically for all the complex
solutions in the sector, which we call two-string solutions. We show that in a
region of $N$ and $\Delta$ the number of two-string solutions is by two larger
than the number due to the string hypothesis, i.e., an extra pair of
two-strings appears. We determine it exactly and also such regions where $m$
two-string solutions collapse for any positive integers $m$. We illustrate the
extra and standard two-string solutions numerically. In the sector we show that
the string deviations are exponentially small with respect to $N$ if $N$ is
large. We argue that for any finite solution of the spin-1/2 XXX chain there is
such a solution of the spin-1/2 XXZ chain that has the same quantum numbers in
common with the XXX solution.
