Darboux Integrability via Singularities of Invariant Curves at Infinity

We extend the computation of the invariant $\eta(\omega,C,a)$ defined in arXiv:2409.01751 to special points on the line at infinity and show that, as in the affine case, its value is determined purely by the geometry of the integral curve C. By incorporating points at infinity, the invariant $\eta$ yields effective geometric criteria that certify Darboux integrability in cases not covered by affine data alone. As an application we construct six new codimension-11 components of the degree-3 center variety