It is likely that the phrase "cooperation in Serbia of the organs of the imperial and royal government in the suppression of the subversive movement directed against the integrity of the monarchy" refers to a historical context, specifically the collaboration between the Serbian government and the Austro-Hungarian Empire in the early 20th century. At that time, the stability and integrity of the Austro-Hungarian Empire were threatened by a number of subversive activities and revolutions throughout the Balkans. The Balkan nation of Serbia was considered to be a possible source of funding for these clandestine movements.
In order to stop this from happening, the Austro-Hungarian Empire enlisted the Serbian government's assistance in putting an end to any subversive initiatives that would jeopardize the Empire's stability. This collaboration probably entailed intelligence sharing, coordinating law enforcement efforts, and perhaps even military cooperation. The phrase implies that the various Austro-Hungarian government agencies collaborated to accomplish this goal and that the Serbian government participated in the suppression of the Balkan subversive activities.