Based on the results of earlier studies on the perturbations of the Schwarzschild and the Reissner–Nordström black holes, it is shown that there exists a very general class of potential pairs (V(+) and V(-) which yield the same reflexion and transmission coefficients. It is further shown that these potentials, V(+) and V(-), satisfy an infinite hierarchy of integral equalities which are, formally, the same as the conserved quantities allowed by the Korteweg–deVries equation.
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S. Chandrasekhar (1980) studied this question.