Puplic address key.
I've never heard of a Public address key.
Are you talking about a public key, or are you talking about a Bitcoin Address? They are NOT the same thing.
I mean that one private key (A) for addresses (AA) and (BB).
where AA and BB compressed.
There are several different types of Bitcoin Addresses. There are P2PKH addresses (which start with a '1'). There are P2SH addresses (which start with a '3'). There are P2WPKH addresses (which start with 'bc1').
About public key.
This is impossible.