It has been recently shown that an observable that identifies all pure states
of a d-dimensional quantum system has minimally 4d-4 outcomes or slightly less
(the exact number depending on the dimension d). However, no simple
construction of this type of observable with minimal number of outcomes is
known. In this work we investigate the possibility to have a covariant
observable that identifies all pure states and has minimal number of outcomes
for this purpose. It is shown that the existence of these kind of observables
depends on the dimension of the Hilbert space. The fact that these kind of
observables fail to exist in some dimensions indicates that the dual pair of
observables -- pure states lacks the symmetry that the dual pair of observables
-- states has