Special equivalent key in symmetric encryption Let $k$ and $k'$ be two keys of symmetric encryption such that for some $m$ we have $ Enc_k(m)=Enc_{k'}(m)$. Is it possible to exist a plain text $m'$ such that $Enc_k(m') \neq Enc_{k'}(m')$.

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 permite 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.