After I have restored the possibility for users to authenticate on my website using their Google accounts, I decided to give a try to the TweetBook plugin for Redmine, which lets authenticate users with their Facebook, Twitter and GitHub accounts. Unfortunately, it appeared to be more complicated, than I expected, and I had to fork and fix this plugin to be able to use it (the original plugin was created by Sandeep Kumar).
Eventually, you can login into my website using your Facebook, Twitter or GitHub account. Also, you can login using you Google account, what was already announced in a separate news.