  The closed relativistic string carrying a point-like mass in the space with
nontrivial geometry is considered. For rotational states of this system
(resulting in non-trivial Regge trajectories) the stability problem is solved.
It was shown that rotations of the folded string with the massive point placed
at the rotational center are stable (with respect to small disturbances) if the
mass exceeds some critical value: $m>m_{cr}$. But these rotational states are
unstable in the opposite case $m<m_{cr}$. We can treat this effect as the
spontaneous symmetry breaking for the string state. Other classes of rotational
motions of this system have appeared to be stable. These results were obtained
both in numerical experiments and the analytical investigation of small
disturbances for the rotational states.
