  We solve the discrete Hirota equations (Kirillov-Reshetikhin Q-systems) for
$A_r$, and their analogue for $D_r$, for the cases where the second variable
ranges over either a finite set or over all integers. Until now only special
solutions were known. We find all solutions for which no component vanishes, as
required in the known applications. As an introduction we present the known
solution where the second variable ranges over the natural numbers.
