How can I safely revert libglapi-mesa version?

Currently I cannot install some software.

sudo apt-get install vlc gives

Reading package lists... Done
Building dependency tree       
Reading state information... Done
Some packages could not be installed. This may mean that you have
requested an impossible situation or if you are using the unstable
distribution that some required packages have not yet been created
or been moved out of Incoming.
The following information may help to resolve the situation:

The following packages have unmet dependencies.
 vlc : Depends: libgles1-mesa (>= 7.8.1) but it is not going to be installed or
                libgles1
       Depends: libgles2-mesa (>= 7.8.1) but it is not going to be installed or
                libgles2
E: Unable to correct problems, you have held broken packages.

Trying to install libgles via Syanptic gives

Which when checking in Syanptic is clear:

Trying to force a specific version back down to 10.5.2 seems to indicate it's going to install a lot of my things which I have configured nicely:

Obviously I don't want to do this, but I would like to be able to use this PC again.

They don't appear in the to be installed list:

grep -r 'ricotz' /etc/apt/ gives:

grep: /etc/apt/trustdb.gpg: Permission denied

sudo grep -r 'ricotz' /etc/apt/ gives no output

sudo apt-cache policy libglapi-mesa gives:

libglapi-mesa:
  Installed: 10.6.0~git20150423.125574d1-0ubuntu0ricotz~utopic
  Candidate: 10.6.0~git20150423.125574d1-0ubuntu0ricotz~utopic
  Version table:
 *** 10.6.0~git20150423.125574d1-0ubuntu0ricotz~utopic 0
        100 /var/lib/dpkg/status
     10.5.2-0ubuntu1 0
        500 http://gb.archive.ubuntu.com/ubuntu/ vivid/main i386 Packages

sudo apt-get install libglapi-mesa=10.5.2-0ubuntu1 gives:

Reading package lists... Done
Building dependency tree       
Reading state information... Done
Some packages could not be installed. This may mean that you have
requested an impossible situation or if you are using the unstable
distribution that some required packages have not yet been created
or been moved out of Incoming.
The following information may help to resolve the situation:

The following packages have unmet dependencies.
 monodoc-browser : Depends: libwebkit1.1-cil (>= 0.3) but it is not going to be installed
E: Error, pkgProblemResolver::Resolve generated breaks, this may be caused by held packages.

sudo apt-cache policy libwebkit1.1.cil gives:

libwebkit1.1-cil:
  Installed: 0.3-6
  Candidate: 0.3-6
  Version table:
 *** 0.3-6 0
        500 http://gb.archive.ubuntu.com/ubuntu/ vivid/main i386 Packages
        100 /var/lib/dpkg/status

Steps from A.B.'s answer:

nikolai@dante:~$ sudo apt-get autoremove
[sudo] password for nikolai: 
Reading package lists... Done
Building dependency tree       
Reading state information... Done
0 to upgrade, 0 to newly install, 0 to remove and 0 not to upgrade.
nikolai@dante:~$ sudo apt-get install -f
Reading package lists... Done
Building dependency tree       
Reading state information... Done
0 to upgrade, 0 to newly install, 0 to remove and 0 not to upgrade.
nikolai@dante:~$ sudo apt-get install libglapi-mesi=10.5.2-0ubuntu1
Reading package lists... Done
Building dependency tree       
Reading state information... Done
E: Unable to locate package libglapi-mesi
nikolai@dante:~$ sudo apt-get install libglapi-mesa=10.5.2-0ubuntu1
Reading package lists... Done
Building dependency tree       
Reading state information... Done
Some packages could not be installed. This may mean that you have
requested an impossible situation or if you are using the unstable
distribution that some required packages have not yet been created
or been moved out of Incoming.
The following information may help to resolve the situation:

The following packages have unmet dependencies.
 monodoc-browser : Depends: libwebkit1.1-cil (>= 0.3) but it is not going to be installed
E: Error, pkgProblemResolver::Resolve generated breaks, this may be caused by held packages.

Solution 1:

The simple solution could be

Downgrade the package libglapi-mesa via

    sudo apt-get install libglapi-mesa=10.5.2-0ubuntu1

You have installed version 10.6.0~git20150423.125574d1-0ubuntu0ricotz~utopic from a PAA, that's not longer exist in your system or manually via dpkg -i or an other tool.

But your system is pretty broken.

Work in progress...

In this case, there is no stubborn sequence of steps. You have to decide and act on their own. I can give you advice only.

Currently we have only Synaptic Package Manager, therefore use it:

• Disable unnecessary PPAs and other repositories. You can re-add them later.

  

  Use Reload

  

• All packages in the category Installed (local or obsolete) should be removed or downgraded

  First try a downgrade. If that's not possible, remove the package. Do this step by step, not all at once. After each removal, check the category Installed (auto removable) and remove the packages also. Be in mind, packages can be re-installed but you should not remove the kernel ;)

  From time to time it's necessary to restart Synaptic. But this assertion comes from my experience and does not have to be true.

  

• Remove packages that can be re-installed

Here are some points from the chat

• Before you restart your system check

  • Is your desktop is still installed?

    apt-cache policy unity
    

  • Is your network manager is installed?

    apt-cache policy network-manager
    

  • Is your graphic installed

    nvidia-settings --version
    

Be in mind and helpful links

• Unity doesn't load, no Launcher, no Dash appears

• What's a good back-up strategy for 1 desktop PC?

• How do I find the reverse dependency of a package?