After the defeat of Napoleon—and even before his final defeat at Waterloo after his escape from exile on Elba—the European leaders planned to turn back the clock and restore the old order on the continent as it had been before 1789. For several reasons, it was impossible to do this. All of Europe had begun to be transformed by Enlightenment values. This transformation had occurred in different ways in each of the three countries named in your question, Austria, Prussia, and Russia.
In the German-speaking countries, the occupation by French forces had discredited the old order and its systems of government, and Napoleon had abolished the Holy Roman Empire. The Germans could no longer maintain the illusion that they had an "empire" that was somehow the heir of ancient Roman power—if they had ever believed this to begin with. Of Austria and Prussia, the latter was now the rising power. There was also a new spirit of German nationalism, a reaction against Napoleon's having defeated the German countries and made client states out of them. German nationalistic feeling paradoxically made it difficult for the individual princes of the former Holy Roman Empire (including those of Austria and even the ascendant Prussia) to retain their authority and absolutist systems.
The situation in Russia was somewhat different. Russia had been devastated in 1812, but not defeated. The Russians regarded themselves as the one country that had saved all of Europe from Napoleon, by wiping out his army and starting the breakup of his empire in 1812–13. The Romanov dynasty seemingly had more credibility than ever before within Russia and in Europe overall. But even in Russia, the army leaders and much of the upper class had been influenced by Enlightenment thinking. The Decembrist revolt in 1825 showed that the Czarist regime and its old values did not have the genuine support of Russian elites.
You might want to consider what the one common thread, in all of the above issues, would be among the three countries in question. It would also be good to investigate how the difficulties the authorities had in restoring conservative values led to the next European-wide cataclysm one hundred years later, in 1914.