Symmetries, Integrability and Exact Solutions for Nonlinear Systems

The paper intends to offer a general overview on what the concept of integrability means for a nonlinear dynamical system and how the symmetry method can be applied for approaching it. After a general part where key problems as direct and indirect symmetry method or optimal system of solutions are tackled out, in the second part of the lecture two concrete models of nonlinear dynamical systems are effectively studied in order to illustrate how the procedure is working out. The two models are the 2D Ricci flow model coming from the general relativity and the 2D convective-diffusion equation . Part of the results, especially concerning the optimal systems of solutions, are new ones. Keywords: Lie symmetries, invariants, similarity reduction.