Has AES some keys which are related to each other? e.g. $\forall m: AES(AES(m,k1),k2)=AES(AES(m,k2),k1)$ or $AES^n(m,k1)=AES(m,k2)$. How to find them?

Do $\exists $ keys $k_1, k_2$ which given any (128-bit) message $m$ are related to each other by being

• commutative to each other with $AES(AES(m,k_1),k_2)=AES(AES(m,k_2),k_1)$ $\forall m$ and generally $AES(m,k_1) \not= AES(m,k_2)$
• or $k_2$ is equal to applying $n$-times $k_1$ with $AES^n(m,k_1)=AES(m,k_2)$ with $AES^n(m,k_1) = AES(AES^{n-1}(m,k_1),k_1)$, and $k_1 \not= k_2, n > 1$,
  target size: $2^5 \ge n \le 2^{64}$ to maintain $n<\sqrt{l}$ in $ \arg\underset l\min AES^l(m,k) = m$ for most $m$,
  for adequate security $n<2^{32}$

AES operates in ECB-mode.
$AES(m,k)$ means applying AES encryption on (128-bit) message with key $k$

If some keys like those exist is there a way to find them?

Statistically speaking such keys might exist for some $m$ but chances are too small to be valid for $\forall m$ unless the inner structure of AES allows some relation in between keys.

bonus:
If such keys can not exists for AES does a similar relation of keys exist for AES or is there some alternative block cipher which can offer such a key relation?