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.

中文翻译：

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奇异曲面上全同叶的不变曲线

C. Camacho和P. Sad的Separatrix定理说，当叶面下面的表面光滑或奇异时，至少有一个不变曲线（separatrix）穿过全同叶状胚芽的奇异性。分辨率表面奇异性的对偶图是一棵树。在某些假设下，即使表面奇异点的分辨率对偶图不是一棵树，我们也能获得分离线的存在。有必要要求额外的叶状条件，即减少奇异点时不存在鞍形结。