The problem of the centre for cubic systems with two parallel invariant straight lines and one invariant conic
For the cubic differential system \[ \begin{aligned} \dot x &= y+ ax^2+ cxy+ fy^2+ kx^3+ mx^2y+ pxy^2+ ry^3,\\ \dot y &= -(x+ gx^2+ dxy+ by^2+ sx^3+ gx^2y+ nxy^2+ ly^3)\end{aligned} \] with two parallel invariant straight lines and at least one invariant conic it is proved that a singular point with pure imaginary eigenvalues is a center if and only if the first three Lyapunov quantities \(L_j\), \(j= 1,2,3\), vanish.
scientific article; zbMATH DE number 5562674