Model equation of the theory of solitons

We consider the hierarchy of integrable $(1+2)$-dimensional equations related to the Lie algebra of vector fields on the line. We construct solutions in quadratures that contain $n$ arbitrary functions of a single argument. A simple equation for the generating function of the hierarchy, which determines the dynamics in negative times and finds applications to second-order spectral problems, is of main interest. Considering its polynomial solutions under the condition that the corresponding potential is regular allows developing a rather general theory of integrable $(1+1)$-dimensional equations.