The partial differential equation u_t/u_x = 1/4 u_(x x x)/u_x - 3/8 u_(x x)^2/u_x^2 + 3/2 (p(u))/u_x^2, where p(u) = 1/4(4u^3 - g_2 u - g_3). The special cases p(u) = (u - e_1)^2(u - e_2) and p(u) = u^3 can be reduced to the Korteweg-de Vries equation by a change of variables.
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