Decay of single-K-shell-vacancy states of xenon atoms by the emission of photon pairs that continuously share the transition energy was studied. The vacancy states were generated by electron-capture decay in $^{131}\mathrm{Cs}$. A pair of high-purity germanium detectors and a fast-slow coincidence system with a (128\ifmmode\times\else\texttimes\fi{}512\ifmmode\times\else\texttimes\fi{}512)-channel pulse-height analyzer were used in the measurements. Accurate energy calibration was made by a careful determination of positions of peaks due to crosstalk among the detectors via Ge K x rays. Identification of events due to the two-photon decay is based on the shifts of energy sums. From the numbers of events in the ridges, the relative differential probabilities of 2s\ensuremath{\rightarrow}1s, 3s\ensuremath{\rightarrow}1s, 3d\ensuremath{\rightarrow}1s, and 4sd\ensuremath{\rightarrow}1s two-photon decay were derived. They are compared to the values derived from the results of non-relativistic calculations for hydrogen and hydrogenic ions of Shapiro and Breit [Phys. Rev. 113, 179 (1959)], of Tung, Ye, Salamo, and Chan [Phys. Rev. A 30, 1175 (1984)], and of Florescu [Phys. Rev. A 30, 2441 (1984)], and to the results of the following studies of two-photon decay of single-K-shell-vacancy states of xenon atoms: a calculation based on the theory of Bannett and Freund [Phys. Rev. Lett. 49, 539 (1982); Phys. Rev. A 30, 299 (1984)], a nonrelativistic calculation of Wu and Li [J. Phys. B 21, 1509 (1988)], and relativistic calculations of Mu and Crasemann [Phys. Rev. Lett. 57, 3039 (1986); Phys. Rev. A 38, 4585 (1988)], and of Tong, Li, Kissel, and Pratt [Phys. Rev. A 42, 1442 (1990)]. As expected, no events due to the np\ensuremath{\rightarrow}1s two-photon transitions were observed and upper limits are given.