socialism [ soh-shuh-liz-uhm ]

noun

a theory or system of social organization that advocates the ownership and control of the means of production and distribution, capital, land, etc., by the community as a whole, usually through a centralized government.

monarchy [ mon-er-kee ]

noun, plural mon·ar·chies.

a state or nation in which the supreme power is actually or nominally lodged in a monarch. Compare absolute monarchy, limited monarchy.

A monarchist system needn't be socialist, just by definitions alone.