Recently, a solution for collision-free trajectories in an N particle thermostatted hard-sphere system undergoing homogeneous shear (the so-called "Sllod" equations of motion) led to a kinetic theory of dilute hard-sphere gases under shear. However, a solution for collisions, necessary for a complete theory at higher densities, has been missing. We present an analytic solution to this problem, which provides surprising insights into the mechanical aspects of thermostatting a system in an external field. The equivalence of constant temperature and constant energy ensembles in the thermodynamic limit in equilibrium, the conditions for the nature of heat exchange with the environment (entropy creation and reduction) in the system, and the condition for appearance of the artificial string phase follow from our solution.