Burglary is a department of the LAPD.

The Desk was removed from the final game of L.A. Noire as stated in an interview with Team Bondi's Brendan McNamara. For a while it was believed this desk would be released as DLC, however Rockstar announced in February 2012 that no further DLC or content would be developed for L.A. Noire [1]. It is also possible that the desk was intentionally left out, in order to give the illusion of Cole spending more time with the LAPD. Cole worked this desk some time between working the Traffic Desk and the Homicide Desk from comments made by captain James Donnelly when Cole first starts working the homicide desk.

It can be assumed that when Phelps was working the Burglary Desk his partner was most likely Harold Caldwell. When Caldwell assists Phelps and Roy Earle in the Vice case Manifest Destiny about who was on the manifest, he and Phelps seem to get along together quite well and they are very friendly towards each other.

There is also evidence that it was going to exist in the game, but then taken out. Next to the telephone in Central Police Station, it says LAPD Telophone Directory: Burglary 2802. You can easily read the phone directory in the case "The Studio Secretary Murder".