A two-dimensional model describing the multilayered anisotropic plate deformations is proposed. The plate is assumed to consist of some orthotropic layers with arbitrary orientation of axes relative to the plate frame. The studied multilayered plate is replaced by the equivalent plate composed of a monoclinic material with piecewise elastic modules. An asymptotic solution is constructed for long-wave deformations. This problem was solved earlier in the first approximation; however, the obtained solution is not applicable for the case in which the stiffness of layers differs essentially from each other. The second asymptotic approximation is constructed in the present paper. It takes into account the effects of transversal shear and the normal fibers extension. Some special cases resulting in simple equations are studied in detail. The asymptotic solution error is estimated by comparison with the exact three-dimensional solutions for some test examples.