  Steerability is a characteristic nonlocal trait of quantum states lying in
between entanglement and Bell nonlocality. A given quantum state is considered
to be steerable if it violates a suitably chosen steering inequality. A quantum
state which otherwise satisfies a certain inequality can violate the inequality
under a global change of basis i.e, if the state is transformed by a nonlocal
unitary operation. Intriguingly there are states which preserve their
non-violation(pertaining to the said inequality) under any global unitary
operation. The present work explores the effect of global unitary operations on
the steering ability of a quantum state which live in two qubits. We
characterize such states in terms of a necessary and sufficient condition on
their spectrum. Such states are also characterized in terms of some analytic
characteristics of the set to which they belong. Looking back at steerability
the present work also provides a relation between steerability and quantum
teleportation together with the derivation of a result related to the optimal
violation of steering inequality . An analytic estimation of the size of such
non-violating states in terms of purity is also obtained. Interestingly the
estimation in terms of purity also gives a necessary and sufficient condition
in terms of bloch parameters of the state. Illustrations from some signature
class of quantum states further underscore our observations.
