Solution of the problem of the centre for cubic differential system with three invariant straight lines in generic position

Abstract

For cubic differential system with three invariant straight lines such that not one pair of the lines is parallel and no more than two lines pass through the same point (in generic position) is proved that a singular point with pure imaginary eigenvalues (a weak focus) is a centre if and only if the first seven Liapunov quantitiesV j , $$j = \overline {1,7} $$ vanish.