Democratic system in which the named "ruler" is a symbolic head of state, and power rests in the hands of elected officials.

democracy

monarchy

constitutional monarchy

democratic socialism

Hm, essentially this would be any system in which the head of the government is not the head of state (so any that does not have presidential system).

This could be an elected president (Austria) or hereditary leader (UK).

So the options are not very optimal here: democracy is just a general term, and so is monarchy (but monarchies can be absolute).

socialism is an economic system, so it's not a good answer

I would choose constitutional monarchy, although it's not the only system that follows this rule.

I would say that parliamentary democracy is the best answer (but not available)