  The controllability issue of control-affine systems on smooth manifolds is
one of the main problems in the theory, and it is recently known [Jouan P.
Equivalence of control systems with linear systems on Lie groups and
homogeneous spaces. ESAIM: Control Optim. Calc. Var. 2010, 16, 956-973] that it
might be connected to that of a particular class of systems called linear
control systems on (a homogeneous manifold of) a Lie group. Note that it may
become very complicated to establish the controllability property of systems
evolving on homogeneous spaces of Lie groups whose dynamics are induced by
those of systems in the Lie group under consideration. In fact, even in
low-dimensional certain homogeneous spaces, this is quite a challenging task,
and for this reason, we have classified in [Da Silva, A., Kizil, E., Duman, O.
Linear Control Systems on Homogeneous Spaces of the Heisenberg Group. J. Dyn.
Control Syst. 2023, 29, 2065-2086] as a first goal all linear control systems
on the homogeneous spaces of the 3-dimensional Heisenberg group $\mathbb{H}$
through its closed subgroups $L$ and, in particular, the controllability and
the control sets have been studied for one of the homogeneous spaces
$L\setminus \mathbb{H}$.
  In this paper, we study the controllability and control sets of the induced
linear control systems in the homogeneous spaces left. In particular, we focus
on the singularity of the induced drift vector fields that results in many
cases and subcases to reveal control sets after quite a technical analysis. We
give some nice illustrations to better understand what is going on
geometrically.
