Sivas -
As far as I've been able to determine, the motor specifications are not posted anywhere online, in the manual or anywhere else. Also, the specifications are not readily visible on the motor itself. (At least, not on my LRI.) Only by removing the motor would it be possible to see if the specs are displayed on a hidden part of the motor. And I don't think anyone wants to dismantle a working machine.
Have you contacted the mfr. yet to see if they will sell you a new motor? They seem to sell all the other parts separately. Their contact info is here:
http://www.litter-robot.com/contact-litter-robot.aspx. They do respond promptly to e-mail inquiries.
If you can find a third-party motor that matches the Robot's specs, you will then be faced with attaching mounting hardware in exactly the correct position on the motor. That may or may not be easy, depending on the size and housing of the motor. If the mounting hardware is not attached exactly, the sprocket teeth will not align properly with the globe and you will have just created more problems.
If you cannot, or do not want to purchase a replacement motor from the mfr., another possibility is to have the motor rebuilt. My knowledge of electric motors is limited, but I believe that would entail having it rewound; However, there may be more to it. You might check with shops that specialize in repairing electric motors, and possibly shops that repair automobile alternators. It's a small and simple motor, so perhaps having it rebuilt wouldn't be that expensive.
I live outside Washington, D.C. and there are several dozen shops that advertise repairing electric motors, so there may be a shop near you as well.
Hope that helps. Good luck!
- Jack