Part of the spectrum of the frequencies of free high-frequency vibrations of a uniform anisotropic plate of variable thickness is investigated by the method of asymptotic integration of the three-dimensional dynamic equations of the theory of elasticity. The modes of vibration, which have one or several deformation half-waves in the direction of the plate thickness, are considered. It is assumed that one of the end surfaces of the plate is plane while the other is smooth with a maximum point. The conditions for the existence of modes of vibration localized in the neighbourhood of the maximum point of the plate thickness are found, and approximate expressions are obtained for the frequencies and modes of vibration of this kind.