My players have expressed interest in a PvP murder mystery, where one of them is secretly the murderer. At first I was enthusiastic, but there are several problems I don't know how to get round, and I was wondering if there was a good game system for this already. I have no preference or requirements in terms of setting or genre.
No Lists, Just Guidelines
I cannot provide you with a list of systems - that is contrary to site policy. However, I can still help you with your question in other ways.
First, make sure your players are generally okay with / interested in such a campaign. PvP can be a thing that ruins parties. If players aren't on board, no system will make things work.
Second, whatever system you use, you'll want to avoid extreme transparency of action resolution. E.g. you want to avoid systems where a player openly spends meta-currencies to influence rolls, because that means it's easy to see who wants to make another PC fail.
Third, you want the system and setting to play nice with taking one player aside and handling things separately. This generally favours settings and systems where a lot can be done remotely, such as Shadowrun, Eclipse Phase, Transhuman Space - which, yes, creates a predisposition towards modern or futuristic settings (or at least settings where magic can serve a similar purpose). I'm saying this because I was actually in a campaign where another PC had motives at odds with mine, and we ended up doing some stuff behind each other's back, and I can't imagine such a plot being pulled off in a historical low-tech game.
Fourth, you want the system to have a solid handling of things like detecting lies, reading motives etc. in a way that allows retention of secrecy. Generally check how the system handles evaluating another person's credibility and see if it works in a way that would ruin your premise, either by making detection too easy for the detector, or by revealing info to the player even when the character didn't figure things out.