A note on Liouvillian first integrals and invariant algebraic curves (Q691756)

Consider the planar polynomial system \[ {dx\over dt}= \sum^\ell_{i=0} b_i(x) y^i,\quad{dy\over dt}= \sum^n_{j=0} a_i(x) y^i,\tag{\(*\)} \] where \(x,y\in\mathbb{C}\), \(t\) can be real or complex, \(b_i\) and \(a_j\) are polynomials in \(x\). The authors assume that \((*)\) has a Liouvillian first integral and ask for additional conditions such that \((*)\) has also a finite invariant algebraic curve. They prove that \(n>\ell+1\) is a sufficient condition for that property.