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Lamb waves in plates exhibiting hysteretic quadratic nonlinearity.Self-induced velocity dispersion and absorption in symmetric and antisymmetric Lamb waves.Shift of the zero group velocity points positions in dispersion of finite amplitude Lamb waves.Temporal dynamics of the absorption and frequency of the nonlinear Lamb waves.Stronger nonlinear effects in standing Lamb modes in comparison with the propagating ones.An analytical theory accounting for the influence of hysteretic nonlinearity of micro-inhomogeneous plate material on the Lamb waves near the S1 zero group velocity point is developed. The theory predicts that the main effect of the hysteretic quadratic nonlinearity consists in the modification of the frequency and the induced absorption of the Lamb modes. The effects of the nonlinear self-action in the propagating and standing Lamb waves are expected to be, respectively, nearly twice and three times stronger than those in the plane propagating acoustic waves. The theory is restricted to the simplest hysteretic nonlinearity, which is influencing only one of the Lamé moduli of the materials. However, possible extensions of the theory to the cases of more general hysteretic nonlinearities are discussed as well as the perspectives of its experimental testing. Applications include nondestructive evaluation of micro-inhomogeneous and cracked plates.