Homogeneity of integrability conditions for multi-parametric families of polynomial-non-linear evolution equations

In this paper we consider the integrability conditions for multi-parametric families of polynomial-non-linear evolution equations with arbitrary parameters as coefficients of differential monomials. These conditions are the necessary ones for the existence of higher-order evolutionary symmetries and conservation laws. Their verification forms the basis for one of the most efficient integrability criteria which is valid both for one-component and multi-component quasi-linear evolution equations in one-temporal and one-spatial dimensions. We show that the integrability conditions, being a system of polynomial equations in arbitrary parameters, in the case of evolution equations with uniform rank have non-trivial homogeneity properties. It allows one to use efficiently the Gröbner bases method combined with the special reduction procedure for homogeneous polynomial systems.