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

Minimal covariant observables identifying all pure states