After all, I find the solution:
Comment this line /etc/ssh/ssh_config > # PasswordAuthentication yes
Maybe you can ask your developers why do you use different authentication for backup (that work without this tweak) and restore, logon to staging server and "add a server, type Linux" that do not work?
It was not easy to find solution with this strange behaviour.
Maybe backup is compatible with ChallengeResponseAuthentication and the other part of your application support only PasswordAuthentication? http://superuser.com/questions/161609/c ... config-fil
By the way the best solution for Unix secure connexion on many servers is Public/Private Key-based and you should think actively to support it. Especially with the new Veeam Agent for Linux