fatal: The remote end hung up unexpectedly by connectiong to GitHub - authentication

Question

I've set up my local system for working with GitHub how descrided in the "Generating SSH Keys" How-To.
The key files and the known_hosts have been created, the public key has been added to the GitHub account, and I get the message "Hi ***! You've successfully authenticated, but GitHub does not provide shell access."
But all my attempts to do something, what needs a connection to the server (EDIT: with writing access) -- git clone, git push etc. -- end with an error:
fatal: The remote end hung up unexpectedly
EDIT:
I can clone, but only via HTTP (https://github.com/account/reponame.git) or Git Read-Only (git://github.com/account/reponame.git).
EDIT:
***#**** ~/.ssh
$ ssh-add -l
Could not open a connection to your authentication agent.
Although an ssh-agent daemon (ssh-agent.exe) is running.
Probably that is the problem.

Answer 1

The only instance where that error message pops up, while your public ssh key is correctly recorded by GitHub, is when
you try to clone a repo with a wrong case or an improper ssh url.
It should be:
git clone git#github.com:user/repo.git
you try to clone a repo for which you don't have write access (public repo), or any access (private repo).
From the GitHub page "Which remote URL should I use?"
To use these URLs, you must have write access to a public repo or any access to a private repo.
These URLs will not work with a public repo you do not have write access to.
the private key is passphrase protected and your authentication agent doesn't run properly:
$ ssh-add -l
Could not open a connection to your authentication agent.
For the last case, consider "How to run (git/ssh) authentication agent?" or "ssh-add complains: Could not open a connection to your authentication agent"

Answer 2

It works!
I've started the Pageant (PuTTY SSH authentication agent) and added my PuTTY private key id_rsa.pkk (exported with PuTTYgen (PuTTY Key Generator) from id_rsa) to it. Now I can write to my repos.
A small how-to for everyone, who has the same troubles I had:
0) Set up the local system for working with GitHub, how descrided in the "Generating SSH Keys" How-To.
1) Open PuTTYgen, load/import the (OpenSSH) private key id_rsa and save it (button Save private key) as id_rsa.pkk (in the same folder, where id_rsa is saved, -- ~/.ssh). You need this PuTTY variant of the key for PuTTY authentification with Pageant and (if you use it) TortoiseGit.
2a) Open Pageant (on my computer it's C:\Program Files (x86)\PuTTY\pageant.exe) and add the PuTTY private key. Or just open the PuTTY private key file (id_rsa.pkk) with Pageant.
2b) For TrtoiseGit I had only to set the SSH client (C:\Program Files (x86)\Git\bin\ssh.exe) in the global TrtoiseGit settings.
That's it.
I've got it working, but I don't have a deep understanding of this
thema and cannot provide a low level explanation of the solution and
the causes of this problem (and e.g why it was working in some clients
and wasn't in some others). So feel free to edit/comment this post, if
you can explain all this better.