Finding equations of the fake projective plane $(C18,p=3,\{2I\})$

We find explicit equations of a new pair of fake projective planes, labeled by $(C18,p=3,\{2I\})$ in the Cartwright-Steger classification. Our method involves starting with known equations of a commensurable fake projective plane $(C18,p=3,\mathrm{\varnothing },d_3{D}_3)$ and working through a chain of cyclic covers and quotients to get to the new one.