  We present new exact solutions describing motions of closed segmented strings
in $AdS_3$ in terms of elliptic functions. The existence of analytic
expressions is due to the integrability of the classical equations of motion,
which in our examples reduce to instances of the McMillan map. We also obtain a
discrete evolution rule for the motion in $AdS_3$ of arbitrary bound states of
fundamental strings and D1-branes in the test approximation.
