  We analyze the density of roots of random polynomials where each complex
coefficient is constructed of a random modulus and a fixed, deterministic
phase. The density of roots is shown to possess a singular component only in
the case for which the phases increase linearly with the index of coefficients.
This means that, contrary to earlier belief, eigenvectors of a typical quantum
chaotic system with some antiunitary symmetry will not display a clustering
curve in the stellar representation. Moreover, a class of time-reverse
invariant quantum systems is shown, for which spectra display fluctuations
characteristic of orthogonal ensemble, while eigenvectors confer to predictions
of unitary ensemble.
