Inverse scattering method for the already found integrable nonlinear evolution equation qt-2(1/ √1+q)xxx=0 is presented. A new integrable nonlinear evolution equation rt+(1-r)3rxxx=0, which can be transformed into the above-mentioned equation, is shown to have a singular spiky soliton solution (cusp soliton).
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Wadati et al. (1980) studied this question.