Why the signature of the same message changes in some wallets, but not all? Using the "message signing" feature in the Bitcoin and Dash it's static with the same message. Signing the same message with the same private key will result always in the same signature.

This behavior is not the same in Peercoin or in the old wallet of Bitcoin. In these wallets, the signature changes even if both messages are equal, for me it has some "nonce" inside. 



Example:

Message: xxx

Address: 18ZbyqYYwgEmn5hVMtJpihPeYxsUpCLhr7

Bitcoin 0.8.5-beta:

INbwudFmql3iOH6c1UWylOpgLu5zrrqGrcFCanirz+Zb/nmJkcDXSkug5J0s3Yawb1/GzZKqXVjRHRNVSenRdMI=
II7W5su7EBQ4u1ppSKMVcmy8etnJsSOsvQHZ3j+moJs2P/vjwpqMlZkvOQ+W5odCdmwOz/5wK2T0RDV6aiqaEQs=
H7XNf5of1kmpwvcvLqVKIYc3TY5eN3kIrZWJPlDYhKVZekZ7Xk2qnJWjNDdsODkYtLEYn5p6fJfIC9LR+/rQky0=


Bitcoin 0.14.2:

IBl6SRpJwzGjXTDbO/IuUcHzlbxqbzN+MmeGRLRIAcPjKpQ9XgQG+gBOzo5PcgtBbtDqXhJIKnbUH9vEcQ5CNlM=
IBl6SRpJwzGjXTDbO/IuUcHzlbxqbzN+MmeGRLRIAcPjKpQ9XgQG+gBOzo5PcgtBbtDqXhJIKnbUH9vEcQ5CNlM=
IBl6SRpJwzGjXTDbO/IuUcHzlbxqbzN+MmeGRLRIAcPjKpQ9XgQG+gBOzo5PcgtBbtDqXhJIKnbUH9vEcQ5CNlM=


In the old Bitcoin-qt the same message with the same private key create different signatures. But in the new Bitcoin-qt it's result in the same signature.



What is the difference between this signature method? Have an alternative to continue getting the PK from a signature in the second case? Have name of this specific change?
You're seeing the effect of RFC6979.

When generating a signature, some randomness is needed (called the nonce), as there are multiple valid signatures for the same message/key pair. The algorithm used, ECDSA, is extraordinarily vulnerable to weak randomness. An attacker who knows just a slight bit of information about your nonces can learn your private key after observing many signatures.

However, someone figured out that it doesn't need to actually be random. The only thing required is that it is unpredictable for the attacker. As a result, a standard was created that suggested using basically the hash of your private key and the message as your nonce. That standard is called RFC6979, and Bitcoin Core switched to using it in version 0.10.0. Clones that forked off before that version typically don't use it.

I have no idea why pubkey recovery does not work without deterministic signatures. I think it should work fine.