  In this article, we completely describe the control sets of one-input linear
control systems on solvable, nonnilpotent 3D Lie groups. We show that, if the
restriction of the associate derivation to the nilradical is nontrivial, the
Lie algebra rank condition is enough to assure the existence of a control set
with a nonempty interior. Moreover, such a control set is unique and, up to
conjugations, given as a cylinder of the state space. On the other hand, if
such a restriction is trivial, one can obtain an infinite number of control
sets with empty interiors or even controllability, depending on the group
considered.
