You mean Sulla?

Also it's not a straight plebeian/patrician divide as many of the senatorial elites were of plebeian origin: Crassus and Pompeius for example. These aren't parties devoted to citizen classes but rather about mechanisms to achieve power/status.

A conservative faction as represented by people like Cato the Younger favoured the traditional methods of achieving political power - by climbing through various offices slowly and incrementally (they favoured it because they dominated the senate and offices). Others opted to use the power of the tribune of the plebs to raise their status through proposed legislation that appealed to the urban population. Regardless of their tactics, all involved were the 1% of the 1%, elites from wealthy families - some with ancient lineages and some less ancient (and a tiny few who had no consular pedigree).

I mean if you wanted to say the reason for the conflict between Marius and Sulla was a refusal to accept any other as being the leading man in Rome, then yes you could say the Caesar/Pompeius conflict was a continuation. But was Caesar fighting for the Marian faction and Pompeius fighting for the Sullan faction? No. The reasons for the civil wars was the inability for one great man to accept any other great man to be their equal. Aristocratic competition had broken the social norms that had formerly regulated it and all bets were off.