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We study the existence and properties of stationary solution of ARCH-type equation r t = ζ t σ t , where ζ t are standardized i.i.d. r.v.'s and the conditional variance satisfies an AR(1) equationand real parameters a, γ, b j . The paper extends the model and the results in [5] from the case γ = 0 to the case 0 < γ < 1. We also obtain a new condition for the existence of higher moments of r t which does not include the Rosenthal constant. In the particular case when Q is the square root of a quadratic polynomial, we prove that r t can exhibit a leverage effect and long memory. We also present simulated trajectories and histograms of marginal density of σ t for different values of γ.Denote |µ| p := E|ζ 0 | p (p > 0), µ p := Eζ p 0 (p = 1, 2, . . . ) and let X t := s

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