On an EKE (Encrypted Key Exchange) what would happen if asymmetric key would be symmetric? On EKE where the user sends a password encrypted public key and receives a password encrypted and public key encrypted secret key. What would happen if these asymmetrical keys would become symmetrical?
On EKE where the user sends a password encrypted public key and receives a password encrypted and public key encrypted secret key.


That's not the original meaning of EKE (which was a password encrypted Diffie Hellman exchange), but the above modification makes sense (if perhaps a bit dangerous), and so lets consider what would happen if we replace the public key encryption with a symmetric one.

In your revised scheme, the Client (Alice) would select a random symmetric key $K$, encrypt it with the password to form $E_{password}(K)$ and send that.  Then, the Server (Bob) would decrypt the ciphertext (using his password) to recover $K$.  Bob then picks a secret key $S$, and encrypts it with both the Alice's symmetric key $K$, and the password, forming $E_{password}(E_K(S))$.  Alice then unwraps $S$, and uses it somehow, perhaps to encrypt a message to Bob.

Now, consider someone listening it; they here $E_{password}(K), E_{password}(E_K(S))$, and $E_S(\text{known text})$.

The don't know $password$, however if they have a guess of the password $password'$, they can attempt to decrypt $E_{password}(K)$ with $password'$, coming up with a value $K'$ (and $K = K'$ if his guess was correct, that is, if $password = password'$).  Then, they can attempt to decrypt $E_{password}(E_K(S))$ with $password'$ and $K'$, resulting in $S'$ (and, again, $S = S'$ if his original guess was correct).  He can then attempt to decrypt $E_S(\text{known text})$; if he original guess is correct, then he will decrypt it properly, and so it will be obvious that his guess was correct.

This is entirely passive, and hence if he has a large dictionary of potential passwords, he can run this logic on each password.  If the correct password appeared somewhere in the dictionary, he'll find it (and then be able to listen into the entire conversation).

This doesn't happen with EKE (either version); if Alice encrypts a random public key, even if the correct password would recover it, he can't use it do decrypt Bob's reply.