There are several banks to work with i.e. USB Bank Cyprus, Marfin Laiki Bank, Bank of Piraeus, Bank of Cyprus etc. you may check them out on your own.
You may also have a look at the Caribbean bank account that we offer here on the forum; /forum/f5/caribbean-offshore-bank-account-100-anonymous-banking-secrecy-18458/ Maybe you can open one of each.