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A new integral equation which linearizes the Korteweg-de Vries and Painlev\'e II equations, and is related to the potentials of the Schr\"odinger eigenvalue problem, is presented. This equation allows one to capture a far larger class of solutions than the Gel'fand-Levitan equation, which may be recovered as a special case. As an application this equation, with the aid of the classical theory of singular integral equations, yields a three-parameter family of solutions to the self-similar reduction of Korteweg -de Vries which is related to Painlev\'e II.
Fokas et al. (Mon,) studied this question.