Solution of the problem of the centre for cubic differential system with three invariant straight lines in generic position (Q996110)

Consider the real system of differential equations \[ \dot x= P(x, y),\quad\dot y= Q(x, y),\tag{1} \] where \(P\), \(Q\) are polynoms of degree three. The following theorem is proved: Theorem. Any singular point \((x_0, y_0)\) with pure imaginary eigenvalues of a system (1) with at least three invariant straight lines (real, complex, real or complex) is a centre if and only if the first seven Lyapunov quantities (focus quantities) vanish at this point.