GURPS has rules for melee combat. That's mostly simple, other than a few quibbles I've mentioned earlier (such as questionable scaling for high strengths with melee weaponry). It has rules for shooting, even contact shooting (+4 to hit, +2 to target's dodge; telegraphic attack with a gun/crossbow/etc).
What they do not have rules for is jamming a gun into someone and pulling the trigger. Weapons with bayonets, hearty revolvers that don't have problems functioning when you violently jam them into the unfortunate in a punchy pistolwhip, etc.
Should it be treated as a Dual weapon attack/combination? One attack as a melee weapon, then a second followup attack with the gun?
Does it count as the telegraphic attack, if it's linked in with the melee smack? If so, is the second attack at no penalty due to the +4 and -4 cancelling, with the defender getting a +1 to parry after the -1 from two attacks on the same target?
My inclination is to treat it as an all out attack (double) or dual weapon attack combo, without the bonuses and penalties for telegraphic attack as it happens "simultaneously" with the melee strike.
It's likely to come up in any setting where there are big beasties or angry people that want to get all up in your business, even moreso in a setting such as the Metro or Monster Hunter campaigns where CQB and enemies disregarding your personal space are the norm, rather than the exception.