  The power spectrum analysis of spectral fluctuations in complex wave and
quantum systems has emerged as a useful tool for studying their internal
dynamics. In this paper, we formulate a nonperturbative theory of the power
spectrum for complex systems whose eigenspectra -- not necessarily of the
random-matrix-theory (RMT) type -- posses stationary level spacings. Motivated
by potential applications in quantum chaology, we apply our formalism to
calculate the power spectrum in a tuned circular ensemble of random $N \times
N$ unitary matrices. In the limit of infinite-dimensional matrices, the exact
solution produces a universal, parameter-free formula for the power spectrum,
expressed in terms of a fifth Painlev\'e transcendent. The prediction is
expected to hold universally, at not too low frequencies, for a variety of
quantum systems with completely chaotic classical dynamics and broken
time-reversal symmetry. On the mathematical side, our study brings forward a
conjecture for a double integral identity involving a fifth Painlev\'e
transcendent.
