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Non-Hermitian systems are known for their intriguing topological properties, which underpin various exotic physical phenomena. Exceptional points, in particular, play a pivotal role in fine-tuning these systems for optimal device functionality and material characteristics. These points can give rise to exceptional surfaces with embedded lower-dimensional non-isolated singularities. Here we introduce a topological classification for non-defective intersection lines of exceptional surfaces, where exceptional surfaces intersect transversally. We achieve this classification by constructing a quotient space of an order-parameter space under equivalence relations of eigenstates. We unveil that the fundamental group of these gapless structures is a non-Abelian group on three generators. This classification not only reveals a unique form of non-Hermitian gapless phases featuring a chain of non-defective intersection lines but …