Cubic diferential systems with an invariant straight line of maximal multiplicity

Abstract

In this work the estimation 3n-2 <= Ma(n) <=3n - 1 of maximal algebraic multiplicity Ma(n) of an invariant straight line is obtained for two-dimensional polynomial dierential systems of degree n>=2. In the class of cubic systems (n = 3) we have Ma(3) = 7. Moreover, we prove that if an affine real invariant straight line has multiplicity equal to 1 (respectively, 2,3,4,5,6,7), then the maximal multiplicity of the line at infinity is 7 (respectively, 5,5,5,4,1,1). Each of these cubic systems has a single affine invariant straight line, is Darboux
integrable and their normal forms are given.