A cyclic edge‐cut of a graph G is an edge set, the removal of which separates two cycles. If G has a cyclic edge‐cut, then it is called cyclically separable. We call a cyclically separable graph super cyclically edge‐connected, in short, super‐λ
c
, if the removal of any minimum cyclic edge‐cut results in a component which is a shortest cycle. In Z. Zhang, B. Wang (2011), it is proved that a connected edge‐transitive graph is super‐λ
c
if either G is cubic with girth at least 7 or G has minimum degree at least 4 and girth at least 6, and the authors also conjectured that a connected graph which is both vertex‐transitive and edge‐transitive is always super cyclically edge‐connected. In this article, for a λ
c
‐optimal but not super‐λ
c
graph G, all possible λ
c
‐superatoms of G which have non‐empty intersection with other λ
c
‐superatoms are determined. This is then used to give a complete classification of non‐super‐λ
c
edge‐transitive k(k≥3)‐regular graphs.
