  The discovery of high-frequency brightness oscillations in thermonuclear
X-ray bursts from several neutron-star low-mass X-ray binaries has important
implications for the beat frequency model of kilohertz quasi-periodic
brightness oscillations, the propagation of nuclear burning, the structure of
the subsurface magnetic fields in neutron stars, and the equation of state of
high-density matter. These implications depend crucially on whether the
observed frequency is the stellar spin frequency or its first overtone. Here we
report an analysis of five bursts from 4U 1636-536 which exhibit strong
oscillations at approximately 580 Hz. We show that combining the data from the
first 0.75 seconds of each of the five bursts yields a signal at 290 Hz that is
significant at the $4\times 10^{-5}$ level when the number of trials is taken
into account. This strongly indicates that 290 Hz is the spin frequency of this
neutron star and that 580 Hz is its first overtone, in agreement with other
arguments about this source but in contrast to suggestions in the literature
that 580 Hz is the true spin frequency. The method used here, which is an
algorithm for combining time series data from the five bursts so that the
phases of the 580 Hz oscillations are aligned, may be used in any source to
search for weak oscillations that have frequencies related in a definite way to
the frequency of a strong oscillation.
