In recent years, there has been a growing interest in the study of complex systems, entanglement, and their behaviors. This interest has been fueled by significant developments in understanding the quantum properties of dynamical systems and in quantum computing. Additionally, the progress made in the field of holographic correspondence has contributed to this heightened interest. In this brief note, we will focus on the concept of Krylov complexity as an efficient method for quantitatively describing the growth of operators within a theoretical framework, specifically concerning a special basis. This basis is generated through iteratively nested commutators between the Hamiltonian and the operator under consideration. Finally, we consider links between Krylov complexity, Painlevé equations and Toda equations.