  In this paper, we study non-wandering homeomorphisms of the two torus in the
identity homotopy class, whose rotation sets are non-trivial line segments from
$(0,0)$ to some totally irrational vector $(\alpha,\beta)$. We show this
rotation set is in fact a non-generic phenomenon for any $C^r$ diffeomorphisms,
with $r \geq 1$. When such a rotation set does happen, assuming several natural
conditions that are generically satisfied in the area-preserving world, we give
a clearer description of its rotational behavior. More precisely, the dynamics
admits bounded deviation along the direction $-(\alpha,\beta)$ in the lift, and
the rotation set is locked inside an arbitrarily small cone with respect to
small $C^0$-perturbations of the dynamics. On the other hand, for any
non-wandering homeomorphism $f$ with this kind of rotation set, we also present
a perturbation scheme in order for the rotation set to be eaten by rotation
sets of nearby dynamics, in the sense that the later set has non-empty interior
and contains the former one. These two flavors interplay and share the common
goal of understanding the stability/instability properties of this kind of
rotation set.
