Summary: The author considers a pair of coupled Painlevé equations arising as a scaling similarity reduction of the Hirota-Satsuma system of partial differential equations. Bäcklund transformations constructed in a previous work are presented explicitly as discrete shifts in a two-dimensional parameter space. \(\tau\)-functions derived from a Hamiltonian description are also presented, which satisfy multilinear lattice equations built from Hirota bilinear operators, and these are used to calculate polynomial \(\tau\)-functions for rational solutions.