The Separatrix Theorem of C. Camacho and P. Sad says that there exists at least one invariant curve (separatrix) passing through the singularity of a germ of holomorphic foliation on complex surface, when the surface underlying the foliation is smooth or when it is singular and the dual graph of resolution surface singularity is a tree. Under some assumptions, we obtain existence of separatrix even when the resolution dual graph of the surface singular point is not a tree. It will be necessary to require an extra condition of the foliation, namely, absence of saddle-node in its reduction of singularities.
"Invariant curves for holomorphic foliations on singular surfaces." Ark. Mat. 58 (1) 179 - 195, April 2020. https://doi.org/10.4310/ARKIV.2020.v58.n1.a11