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In this work we present a scheme for construction of solutions forevolutionary PDEs of some polynomial types q'_t = P(q,q'_x,...), where P is apolynomial in a finite number of variables. This scheme is a generalization ofthe existing technique for solution of completely integrable PDEs using InverseScattering of the Sturm-Liouville differential equation. The KdV equation q'_t= - 3/2 q q'_x + 1/4 q"'_{xxx} is a special case, corresponding to type 1evolutionary equations. We present a complete solution of type 0, and present aKdV hierarchy corresponding to infinite number of polynomial evolutionaryequations rather for \beta = 1/2 \int_0^x q(y,t)dy then for q(x,t) itself, ofthe form  \beta'_t = i^n b_n(\beta_x'), where b_0 = -1/4 \beta"'_{xxx} + 3/2(\beta'_x)^2 corresponds to the KdV equation and  4 (b_{n+1})'_x = -i (b_n)_{xxx}"' + 4i (\beta'_xb_n)'_x. Soliton solutions(i.e. involving pure exponents only) are presented for each such evolutionaryequation, demonstrating a "simplicity" of the solutions construction.

On completely integrable polynomial PDEs arising from Sturm-Liouville differential equation using evolutionary vessels. KdV Hierarchy