"ssh -i <identity file name>" allows you to use a different identity
than the default ~/.ssh/id_[d|r]sa" file.
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Solved. The problem was that ssh-copy-id didn't copied rsa.pub. Pasted it manually to .ssh/authorized_keys on server side. Now everything works like a charm.
Thanks for attention!
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-- Spandex is a privilege, not a right.