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An analytic solution of the problem concerning the frequencies and shapes of free axially symmetric oscillations of a truncated spherical sector filled with an ideal compressible fluid is constructed. A spherical wall of smaller radius and the radial wall of the sector are absolutely rigid. A thin elastic shell whose edge is clamped in the radial wall is located on a spherical boundary of larger radius. The outer surface of the shell borders vacuum. The phenomenon of an anomalous decrease in the fundamental frequency as the spherical walls approach each other is discovered. An approximate formula for determination of the lowest fundamental frequencies, which are approximately proportional to the square root of the difference of the radii of spherical walls for small values of this difference, is constructed and tested numerically. Bibliography: 13 titles.