Centers of a cubic system with an invariant line (Q376351)

This paper concerns the family of ordinary differential equations of the form \[ \dot x = y + R(x,y), \dot y = -x + S(x,y)\tag{1} \] in which \(R\) and \(S\) are cubic polynomials with coefficients in \({\mathbb C}\) such that \(S\) has no linear terms and \(y + R(x, y)\) has the form \((1 + a x)(y + b x^2 + c x y + d y^2)\), i.e., the phase portrait contains an invariant line. Let \(\lambda\) denote the parameter composed of the eleven coefficients of \(R\) and \(S\). The author defines the origin to be a center for a system of the form (1) if the system admits a local analytic first integral of the form \(x^2 + y^2 + \cdots\). The set of elements in the parameter space \({\mathbb C}^{11}\) for which this is true for the corresponding system (1) of ordinary differential equations forms an affine variety \(V\) in \({\mathbb C}^{11}\). The author identifies three ideals \(I_j\), \(j = 1, 2, 3\), in \({\mathbb C}[\lambda]\), the ring of polynomials in the coefficients of (1) with coefficients in \({\mathbb C}\), such that the variety \({\mathbf V}(I_j)\) of each (the subset of \({\mathbb C}^{11}\) on which all elements of \(I_j\) vanish) lies in the center variety \(V\). In each case the underlying system admits an integrating factor.