Toward the Classification of Scalar Nonpolynomial Evolution Equations: Polynomiality in Top Three Derivatives

We prove that arbitrary (nonpolynomial) scalar evolution equations of order m >= 7, that are integrable in the sense of admitting the canonical conserved densities (1), (2), and (3) introduced in [1], are polynomial in the derivatives u(m- i) for i = 0, 1, 2. We also introduce a grading in the algebra of polynomials in u(k) withk >= m - 2 over the ring of functions in x, t, u, ... , u(m-3) and show that integrable equations are scale homogeneous with respect to this grading.