  Linear perturbations for a one parameter family of stationary, closed
Nambu-Goto strings winding around a flat torus in the five-dimensional
Schwarzschild spacetime have been studied. It has been shown that this problem
is solvable in the sense that frequency spectra and perturbation modes can be
expressed only with arithmetic operations and radicals. It has been proven that
the Nambu-Goto strings belonging to this family are always unstable, no matter
how they are located at almost flat region distant from the event horizon.
