A cyclic edge-cut of a connected graph $$G$$ G is an edge set, the removal of which separates two cycles. If $$G$$ G has a cyclic edge-cut, then it is called cyclically separable. For a cyclically separable graph $$G$$ G , the cyclic edge connectivity of a graph $$G$$ G , denoted by $$\lambda _c(G)$$ λ c ( G ) , is the minimum cardinality over all cyclic edge cuts. Let $$X$$ X be a non-empty proper subset of $$V(G)$$ V ( G ) . If $$[X,\overline{X}]=\{xy\in E(G)\ |\ x\in X, y\in \overline{X}\}$$ [ X , X ¯ ] = { x y ∈ E ( G ) | x ∈ X , y ∈ X ¯ } is a minimum cyclic edge cut of $$G$$ G , then $$X$$ X is called a $$\lambda _c$$ λ c -fragment of $$G$$ G . A $$\lambda _c$$ λ c -fragment with minimum cardinality is called a $$\lambda _c$$ λ c -atom. Let $$G$$ G be a $$k (k\ge 3)$$ k ( k ≥ 3 ) -regular cyclically separable graph with $$\lambda _c(G)<g(k-2)$$ λ c ( G ) < g ( k - 2 ) , where $$g$$ g is the girth of $$G$$ G . A combination of the results of Nedela and Skoviera (Math Slovaca 45:481–499, 1995) and Xu and Liu (Australas J Combin 30:41–49, 2004) gives that if $$k\ne 5$$ k ≠ 5 then any two distinct $$\lambda _c$$ λ c -atoms of $$G$$ G are disjoint. The remaining case of $$k=5$$ k = 5 is considered in this paper, and a new proof for Nedela and Škoviera’s result is also given. As a result, we obtain the following result. If $$X$$ X and $$X'$$ X ′ are two distinct $$\lambda _c$$ λ c -atoms of $$G$$ G such that $$X\cap X'\ne \emptyset $$ X ∩ X ′ ≠ ∅ , then $$(k,g)=(5,3)$$ ( k , g ) = ( 5 , 3 ) and $$G[X]\cong K_4$$ G [ X ] ≅ K 4 . As corollaries, several previous results are easily obtained.