The proposed research includes the following two main directions:(1) Using methods and tools of algebraic geometry and computational algebra, we study the integrability of the nonlinear dynamical systems. We focus on finding the varieties of integrability of dynamical systems with the emphasize on the higher dimensional systems. Our approaches are based on combining symbolic computations with methods of the theory of integrability of dynamical systems. We then study bifurcations of limit cycles and critical periods arising after perturbations of higher dimensional integrable systems of differential equation. Furthermore, we intend to study the problem of isochronicity (which is equivalent to the problem of linearization). (2) We propose to study the global topological linearization, linearization of integral manifold, smooth linearization with the emphasis on the study of the linearization of non-autonomous systems when the nonlinear term is unbounded or linear system does not possess exponential dichotomy (in critical state). Up to now there are few results concerning the linearization