  Detection of a planetary ring of exoplanets remains as one of the most
attractive but challenging goals in the field. We present a methodology of a
systematic search for exoplanetary rings via transit photometry of long-period
planets. The methodology relies on a precise integration scheme we develop to
compute a transit light curve of a ringed planet. We apply the methodology to
89 long-period planet candidates from the Kepler data so as to estimate, and/or
set upper limits on, the parameters of possible rings. While a majority of our
samples do not have a sufficiently good signal-to-noise ratio for meaningful
constraints on ring parameters, we find that six systems with a higher
signal-to-noise ratio are inconsistent with the presence of a ring larger than
1.5 times the planetary radius assuming a grazing orbit and a tilted ring.
Furthermore, we identify five preliminary candidate systems whose light curves
exhibit ring-like features. After removing four false positives due to the
contamination from nearby stars, we identify KIC 10403228 as a reasonable
candidate for a ringed planet. A systematic parameter fit of its light curve
with a ringed planet model indicates two possible solutions corresponding to a
Saturn-like planet with a tilted ring. There also remain other two possible
scenarios accounting for the data; a circumstellar disk and a hierarchical
triple. Due to large uncertain factors, we cannot choose one specific model
among the three.
