  The measurement of an exoplanet's oblateness and obliquity provides insights
into the planet's internal structure and formation history. Previous work using
small differences in the shape of the transit light curve has been moderately
successful, but was hampered by the small signal and extreme photometric
precision required. The measurement of changes in transit depth, caused by the
spin precession of an oblate planet, was proposed as an alternative method.
Here, we present the first attempt to measure these changes. Using Kepler
photometry, we examined the brown dwarf Kepler-39b and the warm Saturn
Kepler-427b. We could not reliably constrain the oblateness of Kepler-39b. We
find transit depth variations for Kepler-427b at $90.1\%$ significance
($1.65\sigma$) consistent with a precession period of $P_\mathrm{prec} =
5.45^{+0.46}_{-0.37}~\mathrm{years}$ and an oblateness,
$f~=~0.19^{+0.32}_{-0.16}$. This oblateness is comparable to Solar System gas
giants, and would raise questions about the dynamics and tidal synchronization
of Kepler-427b.
