The Illner model is the most general two-velocity discrete model of a Boltzmann equation in one spatial dimension which satisfies an H-theorem. It includes, as particular cases, both the Carleman and the McKean models. ‘‘Solitons’’ (one-dimensional solutions) and ‘‘bisolitons’’ (two-dimensional, space-plus-time, solutions), which are defined as rational fractions, and solutions with one or two exponential variables are determined. The model is treated as a nonintegrable nonlinear one, and from the solitons the possible class of bisolitons is guessed. Two classes of physically acceptable bisolitons are found. The first class is distributions positive only along one semiaxis and identically zero outside. These are interpreted physically by introducing elastic walls plus source or sink terms which become negligible at infinite time. The second class is periodic solutions which can be seen as damped sound waves. Essentially the same tools are used as in a companion paper for the six-velocity Broadwell model, where the two bisoliton classes mentioned above also exist. This suggests that general methods for obtaining nontrivial exact solutions could exist for the hyperbolic semilinear discrete Boltzmann models.