  Optimization of the mean efficiency for unambiguous (or error
free)discrimination among $N$ given linearly independent nonorthogonal states
should be realized in a way to keep the probabilistic quantum mechanical
interpretation. This imposes a condition on a certain matrix to be positive
semidefinite. We reformulated this condition in such a way that the conditioned
optimization problem for the mean efficiency was reduced to finding an
unconditioned maximum of a function defined on a unit $N$-sphere for
equiprobable states and on an $N$-ellipsoid if the states are given with
different probabilities. We established that for equiprobable states a point on
the sphere with equal values of Cartesian coordinates, which we call symmetric
point, plays a special role. Sufficient conditions for a vector set are
formulated for which the mean efficiency for equiprobable states takes its
maximal value at the symmetric point. This set, in particular, includes
previously studied symmetric states. A subset of symmetric states, for which
the optimal measurement corresponds to a POVM requiring a one-dimensional
ancilla space is constructed. We presented our constructions of a POVM suitable
for the ancilla space dimension varying from 1 till $N$ and the Neumark's
extension differing from the existing schemes by the property that it is
straightforwardly applicable to the case when it is desirable to present the
whole space system + ancilla as the tensor product of a two-dimensional ancilla
space and the $N$-dimensional system space.
