  This paper concerns a kinetic model of the thermostated Boltzmann equation
with a linear deformation force described by a constant matrix. The collision
kernel under consideration includes both the Maxwell molecule and general hard
potentials with angular cutoff. We construct the smooth steady solutions via a
perturbation approach when the deformation strength is sufficiently small. The
steady solution is a spatially homogeneous non Maxwellian state and may have
the polynomial tail at large velocities. Moreover, we also establish the long
time asymptotics toward steady states for the Cauchy problem on the
corresponding spatially inhomogeneous equation in torus, which in turn gives
the non-negativity of steady solutions.
