HOMOGENEITY OF INTEGRABILITY CONDITIONS FOR MULTI-PARAMETRIC FAMILIESOF POLYNOMIAL-NON-LINEAR EVOLUTION-EQUATIONS

In this paper we consider the integrability conditions for multi-param etric families of polynomial-non-linear evolution equations with arbit rary parameters as coefficients of differential monomials. These condi tions are the necessary ones for the existence of higher-order evoluti onary symmetries and conservation laws. Their verification forms the b asis for one of the most efficient integrability criteria which is val id both for one-component and multi-component quasi-linear evolution e quations in one-temporal and one-spatial dimensions. We show that the integrability conditions, being a system of polynomial equations in ar bitrary parameters, in the case of evolution equations with uniform ra nk have non-trivial homogeneity properties. It allows one to use effic iently the Grobner bases method combined with the special reduction pr ocedure for homogeneous polynomial systems.