Arnold's seminar March 28, 2000 K.V. Rerikh "Integrability of functional equations defined by birational mappings. (General theory of integration of birational cascades and cascade-flows)II." The talk is about a positive solution of the problem of algebraic integrability of discrete dynamical systems (DDS) or systems of finite difference equations for two complex functions in one complex variable defined by birational plane mappings. The interest to such functional equations (FEs) or DDS originates from many physical models and digitization of ordinary and partial differential equations (ODEs and PDEs). In previous papers the author dealt with non-algebraically integrable functional equations (Physica D 57 (1992), 337-354; Physica D 82 (1995), 60-78; J. of Math. Phys. 39 (1998), 2821-2832; J. of Geometry and Physics, 24 (1998), 265-290). We will give a summary of main results of these papers.