Let $k$ and $k'$ be two keys of symmetric encryption such that for some $m$ we have $\operatorname{Enc}_k(m)=\operatorname{Enc}_{k'}(m)$. Is it possible to exist a plain text $m'$ such that $\operatorname{Enc}_k(m') \neq \operatorname{Enc}_{k'}(m')$.
In fact, is there exist a key which be equivalent only for special plain text?
Yes. Actually it is quite likely that there are quite a few such keys for block ciphers. That said, finding one takes a collision search which may take a lot of time, especially for ciphers with a large block size.
Proving that there are no other messages that permute to the same ciphertext may even be computationally infeasible.
It would not be a good property for a block cipher to have keys that are fully equivalent. It may not directly destroy all trust in the cipher but it would certainly raise a few eyebrows.
