In RCM, there's a dedicated enzyme called "rep" that's encoded usually
next to the origin nick site. In at least some plasmids (those I used as
a baseline for my designs), rep actually covalently bonds to the 5' end
of the DNA as it nicks, "lifting" the DNA as it nicks it to clearly
expose a loose 3' end for extension.
Although rep initially works as a monomer I think, I recall that in
order to work it needs a pair of reps: possibly the nicking process
requires a dimer, but a monomer is bound covalently? But the partner
monomer resumes its role later in finishing synthesis in such a way that
it permanently inactivates both monomers, preventing catalysis of
additional plasmids. I guess that means it's not an "enzyme" as it isn't
a catalyst, but rather is consumed in the reaction.
I had a paper describing this neatly, but I seem to have lost it. Do
some searching on RCM and rep dimers and you should find some good info.